Building control system with heat disturbance estimation and prediction

ABSTRACT

An environmental control system for a building including heating, ventilation, or air conditioning (HVAC) equipment that operates to affect a zone of the building and a controller including a processing circuit. The processing circuit is configured to estimate a thermal resistance between air of the zone and of an external space using values of a temperature of the zone air, a temperature of the external space air, and a heat transfer rate of the HVAC equipment, each value corresponding to a different time step within a time period. The processing circuit is configured to use the thermal resistance, time step specific values of the temperatures, and time step specific values of the heat transfer rate to estimate corresponding values of a heat disturbance. The processing circuit is configured to operate the HVAC equipment using a model-based control technique based on the heat disturbance values.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

The application is a continuation-in-part of U.S. Patent Application No. 16/418,715 filed May 21, 2019, the entirety of which is incorporated herein by reference.

BACKGROUND

The present disclosure relates generally to control systems for buildings. The present disclosure relates more particularly to system identification for controlling building equipment.

System identification refers to the determination of a model that describes a system. For example, system identification may be used to identify a system describing environmental conditions. Because the physical phenomena that govern such systems are often complex, nonlinear, and poorly understood, system identification requires the determination of model parameters based on measured and recorded data from the real system in order to generate an accurate predictive model. However, various issues such as rounding of sensor measurements to a nearest whole number, high-frequency inputs, etc. can affect accuracy of the predictive model if not properly accounted for.

SUMMARY

One implementation of the present disclosure is an environmental control system for a building, according to some embodiments. The system includes heating, ventilation, or air conditioning (HVAC) equipment that operates to affect a variable state or condition of a zone of the building, according to some embodiments. The system includes a controller including a processing circuit, according to some embodiments. The processing circuit is configured to estimate a thermal resistance between air of the zone and air of an external space outside the zone using values of a temperature of the air of the zone, a temperature of the air of the external space, and a heat transfer rate of the HVAC equipment that provides heating or cooling to the zone, according to some embodiments. Each of the values correspond to a different time step within a time period, according to some embodiments. The processing circuit is configured to, for multiple time steps within the time period, use the thermal resistance between the air of the zone and the air of the external space, a time step specific value of the temperature of the air of the zone, a time step specific value of the temperature of the air of the external space, and a time step specific value of the heat transfer rate of the HVAC equipment to estimate a corresponding time step specific value of a heat disturbance, according to some embodiments. The processing circuit is configured to operate the HVAC equipment over the time period using a model-based control technique based on the time step specific values of the heat disturbance, according to some embodiments.

In some embodiments, the processing circuit is configured to perform a system identification process based on the time step specific values of the heat disturbance to identify a predictive model. The predictive model is used in the model-based control technique, according to some embodiments.

In some embodiments, the system identification process includes identifying a scaling parameter that scales the time step specific values of the heat disturbance.

In some embodiments, the processing circuit is configured to identify a Kalman gain and a stochastic model of the heat disturbance based on the time step specific values of the heat disturbance or predicted heat disturbance values based on the time step specific values of the heat disturbance. The Kalman gain and the stochastic model are used in the model-based control technique, according to some embodiments.

In some embodiments, identifying the Kalman gain and the stochastic model includes performing a multi-step system identification process.

In some embodiments, the processing circuit is configured to filter the time step specific values of the heat disturbance through at least one of an anti-spike filter or a smoothing filter.

In some embodiments, the processing circuit is configured to fit the time step specific values of the heat disturbance to at least one of a Gaussian function, a sinusoid function, or a user-defined function.

Another implementation of the present disclosure is a method for operating heating, ventilation, or air conditioning (HVAC) equipment of a building, according to some embodiments. The method includes estimating a thermal resistance between air of a zone of the building and air of an external space outside the zone using values of a temperature of the air of the zone, a temperature of the air of the external space, and a heat transfer rate of the HVAC equipment that provides heating or cooling to the zone, according to some embodiments. Each of the values correspond to a different time step within a time period, according to some embodiments. The method includes, for multiple time steps within the time period, using the thermal resistance between the air of the zone and the air of the external space, a time step specific value of the temperature of the air of the zone, a time step specific value of the temperature of the air of the external space, and a time step specific value of the heat transfer rate of the HVAC equipment to estimate a corresponding time step specific value of a heat disturbance, according to some embodiments. The method includes operating the HVAC equipment over the time period using a model-based control technique based on the time step specific values of the heat disturbance, according to some embodiments.

In some embodiments, the method includes performing a system identification process based on the time step specific values of the heat disturbance to identify a predictive model. The predictive model is used in the model-based control technique, according to some embodiments.

In some embodiments, the system identification process includes identifying a scaling parameter that scales the time step specific values of the heat disturbance.

In some embodiments, the method includes identifying a Kalman gain and a stochastic model of the heat disturbance based on the time step specific values of the heat disturbance or predicted heat disturbance values based on the time step specific values of the heat disturbance. The Kalman gain and the stochastic model used in the model-based control technique, according to some embodiments.

In some embodiments, identifying the Kalman gain and the stochastic model includes performing a multi-step system identification process.

In some embodiments, the method includes filtering the time step specific values of the heat disturbance through at least one of an anti-spike filter or a smoothing filter.

In some embodiments, the method includes fitting the time step specific values of the heat disturbance to at least one of a Gaussian function, a sinusoid function, or a user-defined function.

Another implementation of the present disclosure is a controller for operating heating, ventilation, or air conditioning (HVAC) equipment of a building, according to embodiments. The controller includes one or more processors, according to some embodiments. The controller includes one or more non-transitory computer-readable media storing instructions that, when executed by the one or more processors, cause the one or more processors to perform operations, according to some embodiments. The operations include estimating a thermal resistance between air of a zone of the building and air of an external space outside the zone using values of a temperature of the air of the zone, a temperature of the air of the external space, and a heat transfer rate of the HVAC equipment that provides heating or cooling to the zone, according to some embodiments. Each of the values correspond to a different time step within a time period, according to some embodiments. The operations include, for multiple time steps within the time period, using the thermal resistance between the air of the zone and the air of the external space, a time step specific value of the temperature of the air of the zone, a time step specific value of the temperature of the air of the external space, and a time step specific value of the heat transfer rate of the HVAC equipment to estimate a corresponding time step specific value of a heat disturbance, according to some embodiments. The operations include operating the HVAC equipment over the time period using a model-based control technique based on the time step specific values of the heat disturbance, according to some embodiments.

In some embodiments, the operations include performing a system identification process based on the time step specific values of the heat disturbance to identify a predictive model. The predictive model is used in the model-based control technique, according to some embodiments.

In some embodiments, the system identification process includes identifying a scaling parameter that scales the time step specific values of the heat disturbance.

In some embodiments, the operations include performing a multi-step system identification process based on the time step specific values of the heat disturbance or predicted heat disturbance values based on the time step specific values of the heat disturbance to identify a Kalman gain and a stochastic model of the heat disturbance. The Kalman gain and the stochastic model are used in the model-based control technique, according to some embodiments.

In some embodiments, the operations include filtering the time step specific values of the heat disturbance through at least one of an anti-spike filter or a smoothing filter.

In some embodiments, the operations include fitting the time step specific values of the heat disturbance to at least one of a Gaussian function, a sinusoid function, or a user-defined function.

Those skilled in the art will appreciate that the summary is illustrative only and is not intended to be in any way limiting. Other aspects, inventive features, and advantages of the devices and/or processes described herein, as defined solely by the claims, will become apparent in the detailed description set forth herein and taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE FIGURES

Various objects, aspects, features, and advantages of the disclosure will become more apparent and better understood by referring to the detailed description taken in conjunction with the accompanying drawings, in which like reference characters identify corresponding elements throughout. In the drawings, like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements.

FIG. 1 is a drawing of a building equipped with a HVAC system, according to some embodiments.

FIG. 2 is a block diagram of the building and HVAC system of FIG. 1, according to some embodiments.

FIG. 3 is a circuit-style diagram of a model of the building and HVAC system of FIG. 1, according to some embodiments.

FIG. 4 is a block diagram of a controller for use with the HVAC system of FIG. 1, according to some embodiments.

FIG. 5 is a detailed block diagram of a model identifier of the controller of FIG. 4, according to some embodiments.

FIG. 6 is flowchart of a process for system identification, according to some embodiments.

FIG. 7 is a flowchart of a multi-step ahead prediction error method for use in system identification, according to some embodiments.

FIG. 8 is a visualization useful in illustrating the multi-step ahead prediction error method of FIG. 7, according to some embodiments.

FIG. 9 is graph of an excitation signal used in a cooling experiment to test the controller of FIG. 4, according to some embodiments.

FIG. 10 is a first graph of results of the cooling experiment of FIG. 9, according to an example experiment.

FIG. 11 is a second graph of results of the cooling experiment of FIG. 9, according to an example experiment.

FIG. 12 is a third graph of results of the cooling experiment of FIG. 9, according to an example experiment.

FIG. 13 is a first pair of graphs of results of a heating experiment that tests the controller of FIG. 4, according to an example experiment.

FIG. 14 is a second pair of graphs of results of the heating experiment of FIG. 13, according to an example experiment.

FIG. 15 is a third pair of graphs of results of the heating experiment of FIG. 13, according to an example experiment.

FIG. 16 is a fourth pair of graphs of results of the heating experiment of FIG. 13, according to an example experiment.

FIG. 17 is a first visualization comparing various results of the heating experiment of FIG. 13, according to an example experiment.

FIG. 18 is a second visualization comparing various results of the heating experiment of FIG. 13, according to an example experiment.

FIG. 19 is a flow diagram of a process for estimating historical heat disturbance, according to some embodiments.

FIG. 20 is a flow diagram of a process for predicting a forecasted heat disturbance and performing model predictive control based on the forecasted heat disturbance, according to some embodiments.

FIG. 21 is a flow diagram of a process for determining a deterministic heat disturbance and performing a model predictive control process based on the deterministic heat disturbance, according to some embodiments.

FIG. 22 is a flow diagram of a process for generating a deterministic model of heat disturbance, according to some embodiments.

FIG. 23 is a flow diagram of a process for training a deterministic model of heat disturbance based on an initial profile of the heat disturbance, according to some embodiments.

FIG. 24 is a flow diagram of a process for identifying a state space system for use in a model predictive control (MPC) process, according to some embodiments.

FIG. 25A is a graph of an estimated heat disturbance profile of a large zone group of a commercial building, according to some embodiments.

FIG. 25B is a graph of an estimated heat disturbance profile of a small zone group of a commercial building, according to some embodiments.

FIG. 25C is a graph of an estimated heat disturbance profile of a small zone group of a residential building, according to some embodiments.

FIG. 26 is a graph of a user defined target daily fitting function used for fitting an initial profile of heat disturbance, according to some embodiments.

FIG. 27 is a graph of a regression performed on data representing a correspondence between heat transfer from HVAC equipment and a temperature difference, according to an example experiment.

FIG. 28 is a graph of an approximate heat disturbance profile over time, according to an example experiment.

FIG. 29 is a graph of deterministic heat disturbance predictions scaled by a scaling parameter, according to an example experiment.

FIG. 30 is a set of graphs illustrating results of performing MPC to determine an optimal heating profile that maximizes cost savings while maintaining temperatures comfortable to occupants, according to an example experiment.

DETAILED DESCRIPTION Overview

Referring generally to the FIGURES, systems and methods for using system identification to estimate heat disturbances in building systems with complex sensor measurements and/or complex heating, ventilation, or air conditioning (HVAC) dynamics are shown and described. The systems and method described herein provide improved system models and therefore improved estimations of heat disturbance to more accurately heat/cool a building while optimizing costs related to said heating/cooling.

As described in the present disclosure, heat disturbance (also referred to as ({dot over (Q)}_(other)) can refer to internal heat load/gains due to solar radiation, occupancy, electrical equipment, etc. A deterministic piece of heat disturbance can describe a portion of a total heat disturbance that can be determined based on parameter values and initial conditions of a heat disturbance estimation. A stochastic piece of heat disturbance can describe some inherent randomness in the heat disturbance. Estimating the stochastic piece can be quite difficult, but may be critical for a control system to generate accurate decisions. In performing control processes such as model predictive control (MPC), accurate estimations of the deterministic piece and the stochastic piece of heat disturbance can be critical for generating control decisions that optimize (e.g., reduce) costs, maintain occupant comfort, etc.

In some embodiments, a building zone group thermal model captures crucial system dynamics for MPC and control. The building zone group thermal model can be described by the following differential equations:

${{C_{ia}{\overset{.}{T}}_{ia}} = {{\frac{1}{R_{mi}}\left( {T_{m} - T_{ia}} \right)} + {\frac{1}{R_{oi}}\left( {T_{oa} - T_{ia}} \right)} + {\overset{.}{Q}}_{HVAC} + {\overset{.}{Q}}_{other}}}{{C_{m}{\overset{.}{T}}_{m}} = {\frac{1}{R_{mi}}\left( {T_{ia} - T_{m}} \right)}}$

where T_(ia) is an indoor air temperature, T_(oa) is an outdoor air temperature, T_(m) is a lumped thermal mass temperature, T_(sp) is an indoor air temperature setpoint, C_(m) is a lumped mass thermal capacitance, C_(ia) is an indoor air thermal capacitance, R_(mi) is a thermal resistance between indoor air and thermal mass, R_(oi) is a thermal resistance between indoor air and outdoor air, ({dot over (Q)}_(HAVAC) is a heat provided to and/or removed from a building space by an HVAC system, and {dot over (Q)}_(other) is internal heat load/gains due to solar radiation, occupancy, electrical equipment, etc.

In the above building zone group thermal model, {dot over (Q)}_(HVAC), T_(oa), and {dot over (Q)}_(other) can be inputs to the thermal model. {dot over (Q)}_(HVAC) can be measured and controlled, T_(oa) can be measured but cannot be controlled, and {dot over (Q)}_(other) can be neither measured nor controlled. The states of the above thermal model can be seen as T_(ia), a measured state, and T_(m), a non-measured state. An output to the thermal model can be T_(ia). To obtain values of the resistances and capacitances (i.e., C_(ia), C_(m), R_(mi), and R_(oi)) of the thermal model, a system identification process can be performed. To perform the system identification, data for the building zone can be collected and used to solve for the parameters that provide accurate predictions of the building thermal dynamics.

In some embodiments, the system identification process is performed by a two-step process where a first step puts the building thermal model into a state space form and identifies the resistances and capacitances (i.e., C_(ia), C_(m), R_(mi), and R_(oi)) using {dot over (Q)}_(HVAC) and T_(oa) as inputs. Based on the first step, a measured output T_(ia), can be obtained. In a second step, a disturbance model can be augmented to the system to estimate the heat disturbance values. In general, the augmented model can be given by the following state space representation:

$\begin{bmatrix} {\overset{.}{x}(t)} \\ {\overset{.}{d}(t)} \end{bmatrix} = {{\begin{bmatrix} A_{c} & B_{d} \\ 0 & A_{d} \end{bmatrix}\begin{bmatrix} {x(t)} \\ {d(t)} \end{bmatrix}} + {\begin{bmatrix} B_{c} \\ B_{dd} \end{bmatrix}{u(t)}}}$ ${y(t)} = {{\left\lbrack {C_{c}\ C_{d}} \right\rbrack \begin{bmatrix} {x(t)} \\ {d(t)} \end{bmatrix}} + {D_{c}{u(t)}}}$

where the disturbance model is characterized by the A_(d), B _(d), B_(dd), and C_(d) matrices. Further, the system can be converted into a discrete time model and an observer gain can be identified resulting in the following the closed-loop state estimation system:

$\begin{bmatrix} {\overset{\hat{}}{x}\left( {t + 1} \middle| t \right)} \\ {\overset{\hat{}}{d}\left( {t + 1} \middle| t \right)} \end{bmatrix} = {{\begin{bmatrix} A_{dis} & B_{d_{dis}} \\ 0 & A_{d_{dis}} \end{bmatrix}\begin{bmatrix} {\overset{\hat{}}{x}\left( t \middle| {t - 1} \right)} \\ {\overset{\hat{}}{d}\left( t \middle| {t - 1} \right)} \end{bmatrix}} + {\begin{bmatrix} B_{dis} \\ B_{dd_{dis}} \end{bmatrix}{u(t)}} + {\underset{\underset{= {:{K{(\varphi)}}}}{}}{\begin{bmatrix} {K_{x}(\varphi)} \\ {K_{d}(\varphi)} \end{bmatrix}}\left( {{y(t)} - {\overset{\hat{}}{y}\left( t \middle| {t - 1} \right)}} \right)}}$ $\mspace{20mu} {{\overset{\hat{}}{y}\left( t \middle| {t - 1} \right)} = {{\left\lbrack {C_{dis}\ C_{d_{dis}}} \right\rbrack \begin{bmatrix} {\overset{\hat{}}{x}\left( t \middle| {t - 1} \right)} \\ {\overset{\hat{}}{d}\left( t \middle| {t - 1} \right)} \end{bmatrix}} + {D_{dis}{u(t)}}}}$

where the matrices A_(dis), B_(dis), C_(dis), D_(dis), A_(d) _(dis) , B_(d) _(dis) , B_(dd) _(dis) and C_(d) _(dis) are discrete versions of the above A_(d), B_(d), B_(dd), and C_(d) matrices and the K matrix is the observer gain.

However, augmenting the disturbance model and/or estimating the disturbance values may fail if an inaccurate disturbance model is used, if sensors utilize quantized measurements (i.e., measurements are rounded to a nearest integer), or if high frequency inputs permeate the data. To mitigate these issues, various regressions based on steady state assumptions can be performed to calculate an initial profile of the heat disturbance. The initial profile can be used in conjunction with various measurements of environmental conditions to obtain a model that accurately predicts changes in the heat disturbance over time. Particularly, the model can be used to predict scaled values of the heat disturbance over time and can be used in control processes to operate building equipment to optimize (e.g., reduce) overall costs and maintain occupant comfort. These and other features of the systems and methods are described in detail below.

Building HVAC Systems

Referring to FIG. 1, a perspective view of a building 10 is shown. Building 10 is served by a building management system (BMS). A BMS is, in general, a system of devices configured to control, monitor, and manage equipment in or around a building or building area. A BMS can include, for example, a HVAC system, a security system, a lighting system, a fire alerting system, any other system that is capable of managing building functions or devices, or any combination

The BMS that serves building 10 includes a HVAC system 100. HVAC system 100 can include a plurality of HVAC devices (e.g., heaters, chillers, air handling units, pumps, fans, thermal energy storage, etc.) configured to provide heating, cooling, ventilation, or other services for building 10. For example, HVAC system 100 is shown to include a waterside system 120 and an airside system 130. Waterside system 120 may provide a heated or chilled fluid to an air handling unit of airside system 130. Airside system 130 may use the heated or chilled fluid to heat or cool an airflow provided to building 10.

HVAC system 100 is shown to include a chiller 102, a boiler 104, and a rooftop air handling unit (AHU) 106. Waterside system 120 may use boiler 104 and chiller 102 to heat or cool a working fluid (e.g., water, glycol, etc.) and may circulate the working fluid to AHU 106. In various embodiments, the HVAC devices of waterside system 120 can be located in or around building 10 (as shown in FIG. 1) or at an offsite location such as a central plant (e.g., a chiller plant, a steam plant, a heat plant, etc.). The working fluid can be heated in boiler 104 or cooled in chiller 102, depending on whether heating or cooling is required in building 10. Boiler 104 may add heat to the circulated fluid, for example, by burning a combustible material (e.g., natural gas) or using an electric heating element. Chiller 102 may place the circulated fluid in a heat exchange relationship with another fluid (e.g., a refrigerant) in a heat exchanger (e.g., an evaporator) to absorb heat from the circulated fluid. The working fluid from chiller 102 and/or boiler 104 can be transported to AHU 106 via piping 108.

AHU 106 may place the working fluid in a heat exchange relationship with an airflow passing through AHU 106 (e.g., via one or more stages of cooling coils and/or heating coils). The airflow can be, for example, outside air, return air from within building 10, or a combination of both. AHU 106 may transfer heat between the airflow and the working fluid to provide heating or cooling for the airflow. For example, AHU 106 can include one or more fans or blowers configured to pass the airflow over or through a heat exchanger containing the working fluid. The working fluid may then return to chiller 102 or boiler 104 via piping 110.

Airside system 130 may deliver the airflow supplied by AHU 106 (i.e., the supply airflow) to building 10 via air supply ducts 112 and may provide return air from building 10 to AHU 106 via air return ducts 114. In some embodiments, airside system 130 includes multiple variable air volume (VAV) units 116. For example, airside system 130 is shown to include a separate VAV unit 116 on each floor or zone of building 10. VAV units 116 can include dampers or other flow control elements that can be operated to control an amount of the supply airflow provided to individual zones of building 10. In other embodiments, airside system 130 delivers the supply airflow into one or more zones of building 10 (e.g., via supply ducts 112) without using intermediate VAV units 116 or other flow control elements. AHU 106 can include various sensors (e.g., temperature sensors, pressure sensors, etc.) configured to measure attributes of the supply airflow. AHU 106 may receive input from sensors located within AHU 106 and/or within the building zone and may adjust the flow rate, temperature, or other attributes of the supply airflow through AHU 106 to achieve setpoint conditions for the building zone.

HVAC system 100 thereby provides heating and cooling to the building 10. The building 10 also includes other sources of heat transfer that the indoor air temperature in the building 10. The building mass (e.g., walls, floors, furniture) influences the indoor air temperature in building 10 by storing or transferring heat (e.g., if the indoor air temperature is less than the temperature of the building mass, heat transfers from the building mass to the indoor air). People, electronic devices, other appliances, etc. (“heat load”) also contribute heat to the building 10 through body heat, electrical resistance, etc. Additionally, the outside air temperature impacts the temperature in the building 10 by providing heat to or drawing heat from the building 10.

HVAC System and Model

Referring now to FIG. 2, a block diagram of the HVAC system 100 with building 10 is shown, according to an exemplary embodiment. More particularly, FIG. 2 illustrates the variety of heat transfers that affect the indoor air temperature Tia of the indoor air 201 in zone 200 of building 10. Zone 200 is a room, floor, area, etc. of building 10. In general, the primary goal of the HVAC system 100 is to maintain the indoor air temperature Ea in the zone 200 at or around a desired temperature to facilitate the comfort of occupants of the zone 200 or to meet other needs of the zone 200.

As shown in FIG. 2, the indoor air temperature Tia of the zone 200 has a thermal capacitance Cm. The indoor air temperature Tia is affected by a variety of heat transfers {dot over (Q)} into the zone 200, as described in detail below. It should be understood that although all heat transfers {dot over (Q)} are shown in FIG. 2 as directed into the zone 200, the value of one or more of the heat transfers {dot over (Q)} may be negative, such that heat flows out of the zone 200.

The heat load 202 contributes other heat transfer Qother to the zone 200. The heat load 202 includes the heat added to the zone by occupants (e.g., people, animals) that give off body heat in the zone 200. The heat load 202 also includes computers, lighting, and other electronic devices in the zone 200 that generate heat through electrical resistance, as well as solar irradiance.

The building mass 204 contributes building mass heat transfer {dot over (Q)}_(m) to the zone 200. The building mass 204 includes the physical structures in the building, such as walls, floors, ceilings, furniture, etc., all of which can absorb or give off heat. The building mass 204 has a temperature T_(m) and a lumped mass thermal capacitance C_(m). The resistance of the building mass 204 to exchange heat with the indoor air 201 (e.g., due to insulation, thickness/layers of materials, etc.) may be characterized as mass thermal resistance R_(mi).

The outdoor air 206 contributes outside air heat transfer {dot over (Q)}_(oa) to the zone 200. The outdoor air 206 is the air outside of the building 10 with outdoor air temperature T_(oa). The outdoor air temperature T_(oa) fluctuates with the weather and climate. Barriers between the outdoor air 206 and the indoor air 201 (e.g., walls, closed windows, insulation) create an outdoor-indoor thermal resistance R_(oi) to heat exchange between the outdoor air 206 and the indoor air 201.

The HVAC system 100 also contributes heat to the zone 200, denoted as {dot over (Q)}_(HVAC). The HVAC system 100 includes HVAC equipment 210, controller 212, an indoor air temperature sensor 214 and an outdoor air temperature sensor 216. The HVAC equipment 210 may include the waterside system 120 and airside system 130 of FIG. 1, or other suitable equipment for controllably supplying heating and/or cooling to the zone 200. In general, HVAC equipment 210 is controlled by a controller 212 to provide heating (e.g., positive value of {dot over (Q)}_(HVAC)) or cooling (e.g., a negative value of {dot over (Q)}_(HVAC)) to the zone 200.

The indoor air temperature sensor 214 is located in the zone 200, measures the indoor air temperature T_(ia), and provides the measurement of T_(ia) to the controller 212. The outdoor air temperature sensor 216 is located outside of the building 10, measures the outdoor air temperature T_(oa), and provides the measurement of T_(oa) to the controller 212.

The controller 212 receives the temperature measurements T_(oa) and T_(ia), generates a control signal for the HVAC equipment 210, and transmits the control signal to the HVAC equipment 210. The operation of the controller 212 is discussed in detail below. In general, the controller 212 considers the effects of the heat load 202, building mass 204, and outdoor air 206 on the indoor air 201 in controlling the HVAC equipment 210 to provide a suitable level of {dot over (Q)}_(HVAC). A model of this system for use by the controller 212 is described with reference to FIG. 3.

In the embodiments described herein, the control signal provide to the HVAC equipment 210 by the controller 110 indicates a temperature setpoint T_(sp) for the zone 200. To determine the temperature setpoint T_(sp), the controller 212 assumes that the relationship between the indoor air temperature T_(ia) and the temperature setpoint T_(sp) follows a proportional-integral control law with saturation, represented as:

{dot over (Q)} _(HVAC,j) =K _(p,j)ε_(sp) +K _(i,j)∫E₀ ^(t)ε_(sp)(s)ds   (Eq. A)

ε_(sp) =T _(sp,j) −T _(ia)   (Eq. B)

where j∈{clg, hlg} is the index that is used to denote either heating or cooling mode. Different parameters K_(p,j) and K_(i,j) are needed for the heating and cooling mode. Moreover, the heating and cooling load is constrained to the following set: {dot over (Q)}_(HVAC,j)∈[0, {dot over (Q)}_(cig,max)] for cooling mode (j=clg) and {dot over (Q)}_(HVAC,j)∈[−{dot over (Q)}_(htg,max)0] for heating mode (j=htg). As discussed in detail below with reference to FIG. 4, the controller 212 uses this model in generating a control signal for the HVAC equipment 210.

Referring now to FIG. 3, a circuit-style diagram 300 corresponding to the zone 200 and the various heat transfers {dot over (Q)} of FIG. 2 is shown, according to an exemplary embodiment. In general, the diagram 300 models the zone 200 as a two thermal resistance, two thermal capacitance, control-oriented thermal mass system. This model can be characterized by the following system of linear differential equations, described with reference to FIG. 3 below:

$\begin{matrix} {{C_{ia}{\overset{.}{T}}_{ia}} = {{\frac{1}{R_{mi}}\left( {T_{m} - T_{ia}} \right)} + {\frac{1}{R_{oi}}\left( {T_{oa} - T_{ia}} \right)} - {\overset{.}{Q}}_{HVAC} + {\overset{.}{Q}}_{other}}} & \left( {{Eq}.\mspace{11mu} C} \right) \\ {\mspace{76mu} {{C_{m}{\overset{.}{T}}_{m}} = {\frac{1}{R_{mi}}\left( {T_{ia} - T_{m}} \right)}}} & \left( {{Eq}.\mspace{11mu} D} \right) \end{matrix}$

where the first line (Eq. C) focuses on the indoor air temperature T_(ia), and each term in Eq. C corresponds to a branch of diagram 300 as explained below:

Indoor air node 302 corresponds to the indoor air temperature T_(ia). From indoor air node 302, the model branches in several directions, including down to a ground 304 via a capacitor 306 with a capacitance C_(ia). The capacitor 306 models the ability of the indoor air to absorb or release heat and is associated with the rate of change of the indoor heat transfer {dot over (T)}_(ia). Accordingly, the capacitor 306 enters Eq. C on the left side of the equation as C_(ia){dot over (T)}_(ia).

From indoor air node 302, the diagram 300 also branches left to building mass node 310, which corresponds to the thermal mass temperature T_(m). A resistor 312 with mass thermal resistance R_(mi) separates the indoor air node 302 and the building mass node 310, modeling the heat transfer {dot over (Q)}_(m) from the building mass 204 to the indoor air 201 as

${\frac{1}{R_{mi}}\left( {T_{m} - T_{ia}} \right)}.$

This term is included on the right side of Eq. C above as contributing to the rate of change of the indoor air temperature {dot over (T)}_(ia).

The diagram 300 also branches up from indoor air node 302 to outdoor air node 314. A resistor 316 with outdoor-indoor thermal resistance R_(oi) separates the indoor air node 302 and the outdoor air node 314, modeling the flow heat from the outdoor air 206 to the indoor air 201 as

${\frac{1}{R_{oi}}\left( {T_{oa} - T_{ia}} \right)}.$

This term is also included on the right side of Eq. C above as contributing to the rate of change of the indoor air temperature {dot over (T)}_(ia).

Also from indoor air node 302, the diagram 300 branches right to two {dot over (Q)} sources, namely {dot over (Q)}_(HVAC) and {dot over (Q)}_(other). As mentioned above, {dot over (Q)}_(other) corresponds to heat load 202 and to a variety of sources of energy that contribute to the changes in the indoor air temperature T_(ia). {dot over (Q)}_(other) is not measured or controlled by the HVAC system 100, yet contributes to the rate of change of the indoor air temperature {dot over (T)}_(ia). {dot over (Q)}_(HVAC) is generated and controlled by the HVAC system 100 to manage the indoor air temperature T_(ia). Accordingly, {dot over (Q)}_(HVAC) and {dot over (Q)}_(other) are included on the right side of Eq. C above.

The second differential equation (Eq. D) above focuses on the rate of change {dot over (T)}_(m) in the building mass temperature T. The capacity of the building mass to receive or give off heat is modelled by capacitor 318. Capacitor 318 has lumped mass thermal capacitance C_(m) and is positioned between a ground 304 and the building mass node 310 and regulates the rate of change in the building mass temperature T_(m). Accordingly, the capacitance C_(m) is included on left side of Eq. D. Also branching from the building mass node 310 is resistor 312 leading to indoor air node 302. As mentioned above, this branch accounts for heat transfer {dot over (Q)}_(m) between the building mass 204 and the indoor air 201. Accordingly, the term

$\frac{1}{R_{mi}}\left( {T_{ia} - T_{m}} \right)$

is included on the right side of Eq. D.

As described in detail below, the model represented by diagram 300 is used by the controller 212 in generating a control signal for the HVAC equipment 210. More particularly, the controller 212 uses a state-space representation of the model shown in diagram 300. The state-space representation used by the controller 212 can be derived by incorporating Eq. A and B with Eq. C and D, and writing the resulting system of equations as a linear system of differential equations to get:

$\begin{matrix} {\begin{bmatrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ i \end{bmatrix} = {\begin{bmatrix} {\frac{1}{C_{ia}}\left( {K_{p,j} - \frac{1}{R_{m\; i}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{m\; i}} & \frac{K_{I,j}}{C_{ia}} \\ \frac{1}{C_{m}R_{m\; i}} & {- \frac{1}{C_{m}R_{m\; i}}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}{\quad{{\begin{bmatrix} T_{ia} \\ T_{m} \\ I \end{bmatrix} + {\begin{bmatrix} {- \frac{K_{p,j}}{C_{ia}}} & \frac{1}{C_{ia}R_{oi}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} T_{spj} \\ T_{oa} \end{bmatrix}} + {\begin{bmatrix} \frac{1}{C_{ia}} \\ 0 \\ 0 \end{bmatrix}{\overset{.}{Q}}_{other}}};}}}} & \left( {{Eq}.\mspace{11mu} E} \right) \\ {\mspace{85mu} {{\begin{bmatrix} T_{ia} \\ {\overset{.}{Q}}_{{HVAC},j} \end{bmatrix} = {{\begin{bmatrix} 1 & 0 & 0 \\ {- K_{p,j}} & 0 & K_{I,j} \end{bmatrix}\begin{bmatrix} T_{ia} \\ T_{m} \\ I \end{bmatrix}} + {\begin{bmatrix} 0 & 0 \\ K_{p,j} & 0 \end{bmatrix}\begin{bmatrix} T_{{sp},j} \\ T_{oa} \end{bmatrix}}}};}} & \left( {{Eq}.\mspace{11mu} F} \right) \end{matrix}$

where I represents the integral term ∫₀ ^(t)ε_(sp)(S) ds from Eq. A. The resulting linear system has three states (T_(ia), T_(m), I), two inputs (T_(sp, j), T_(oa)), two outputs (T_(ia), {dot over (Q)}_(HVAC)), and one disturbance {dot over (Q)}_(other). Because {dot over (Q)}_(other) is not measured or controlled, the controller 212 models the disturbance {dot over (Q)}_(other) using an input disturbance model that adds a forth state d to the state space representation. In a more compact form, this linear system of differential equations can be written as:

$\begin{matrix} {{{\overset{.}{x}(t)} = {{{A_{c}(\theta)}{x(t)}} + {{B_{c}(\theta)}{u(t)}}}};} & \left( {{Eq}.\mspace{11mu} G} \right) \\ {{{{y(t)} = {{{C_{c}(\theta)}{x(t)}} + {{D_{c}(\theta)}{u(t)}}}};}{where}{{{A_{c}(\theta)} = \begin{bmatrix} {- \left( {\theta_{1} + \theta_{2} + {\theta_{3}\theta_{4}}} \right)} & \theta_{2} & {\theta_{3}\theta_{4}\theta_{5}} \\ \theta_{6} & {- \theta_{6}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}},{{B_{c}(\theta)} = \begin{bmatrix} {\theta_{3}\theta_{4}} & \theta_{1} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}},{{C_{c}(\theta)} = \begin{bmatrix} 1 & 0 & 0 \\ {- \theta_{4}} & 0 & {\theta_{5}\theta_{4}} \end{bmatrix}},\ {{{D_{c}(\theta)} = \begin{bmatrix} 0 & 0 \\ \theta_{4} & 0 \end{bmatrix}};}}{{\theta_{1} = \frac{1}{C_{ia}R_{oi}}};}{{\theta_{2} = \frac{1}{C_{ia}R_{mi}}};}{{\theta_{3} = \frac{1}{C_{ia}}};}{{\theta_{4} = K_{p}};}{{\theta_{5} = \frac{1}{\tau}};}{{\theta_{6} = \frac{1}{C_{m}R_{mi}}};{and}}{{{\overset{.}{x}(t)} = \begin{bmatrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ i \end{bmatrix}};{{x(t)} = \begin{bmatrix} T_{ia} \\ T_{m} \\ l \end{bmatrix}};{{u(t)} = {\begin{bmatrix} T_{spj} \\ T_{oa} \end{bmatrix}.}}}} & \left( {{Eq}.\mspace{11mu} H} \right) \end{matrix}$

As described in detail below, the controller 212 uses a two-step process to parameterize the system. In the first step, the controller 212 identifies the system parameters θ={θ₁, θ₂, θ₃, θ₄, θ₅, θ₆} (i.e., the values of C_(ia), C_(m), R_(mi), R_(oi), K_(p,j), K_(i,j)). The disturbance state d is then introduced into the model and an Kalman estimator gain is added, such that in the second step the controller 212 identifies the Kalman gain parameters K. In some embodiments, the temperature setpoint T_(sp) is not used as a system input, rather, {dot over (Q)}_(HVAC) is used as a direct input for controller 212 in generating a control signal for the HVAC equipment 210.

As used herein, the term ‘variable’ refers to an item/quantity capable of varying in value over time or with respect to change in some other variable. A “value” as used herein is an instance of that variable at a particular time. A value may be measured or predicted. For example, the temperature setpoint T_(sp) is a variable that changes over time, while T_(sp)(3) is a value that denotes the setpoint at time step 3 (e.g., 68 degrees Fahrenheit). The term “predicted value” as used herein describes a quantity for a particular time step that may vary as a function of one or more parameters.

Controller for HVAC Equipment with System Identification

Referring now to FIG. 4, a detailed diagram of the controller 212 is shown, according to an exemplary embodiment. The controller 212 includes a processing circuit 400 and a communication interface 402. The communication interface 402 is structured to facilitate the exchange of communications (e.g., data, control signals) between the processing circuit 400 and other components of HVAC system 100. As shown in FIG. 4, the communication interface 402 facilitates communication between the processing circuit 400 and the outdoor air temperature sensor 216 and the indoor air temperature sensor 214 to all temperature measurements T_(oa) and T_(ia) to be received by the processing circuit 400. The communication interface 402 also facilitates communication between the processing circuit 400 and the HVAC equipment 210 that allows a control signal (indicated as temperature setpoint T_(sp)) to be transmitted from the processing circuit 400 to the HVAC equipment 210.

The processing circuit 400 is structured to carry out the functions of the controller described herein. The processing circuit 400 includes a processor 404 and a memory 406. The processor 404 may be implemented as a general-purpose processor, an application-specific integrated circuit, one or more field programmable gate arrays, a digital signal processor, a group of processing components, or other suitable electronic processing components. The memory 406, described in detail below, includes one or more memory devices (e.g., RAM, ROM, NVRAM, Flash Memory, hard disk storage) that store data and/or computer code for facilitating at least some of the processes described herein. For example, the memory 406 stores programming logic that, when executed by the processor 404, controls the operation of the controller 212. More particularly, the memory 406 includes a training data generator 408, a training data database 410, a model identifier 412, a model predictive controller 414, and an equipment controller 416. The various generators, databases, identifiers, controllers, etc. of memory 406 may be implemented as any combination of hardware components and machine-readable media included with memory 406.

The equipment controller 416 is configured to generate a temperature setpoint T_(sp) that serves as a control signal for the HVAC equipment 210. The equipment controller receives inputs of the indoor air temperature T_(ia) from the indoor air temperature sensor 214 via the communication interface 402 and {dot over (Q)}_(HVAC) from the model predictive controller 414 (during normal operation) and the training data generator 408 (during a training data generation phase described in detail below). The equipment controller uses T_(ia) and {dot over (Q)}_(HVAC) to generate T_(sp) by solving Eq. A and Eq. B above for T_(sp). The equipment controller 416 then provides the control signal T_(sp) to the HVAC equipment 210 via the communication interface 402.

The model predictive controller 414 determines {dot over (Q)}_(HVAC) based on an identified model and the temperature measurements T_(ia), T_(oa), and provides {dot over (Q)}_(HVAC) to the equipment controller 416. The model predictive controller 414 follows a model predictive control (MPC) approach. The MPC approach involves predicting future system states based on a model of the system, and using those predictions to determine the controllable input to the system (here, {dot over (Q)}_(HVAC)) that bests achieves a control goal (e.g., to maintain the indoor air temperature near a desired temperature). A more accurate model allows the MPC to provide better control based on more accurate predictions. Because the physical phenomena that define the behavior of the system (i.e., of the indoor air 201 in the building 10) are complex, nonlinear, and/or poorly understood, a perfect model derived from first-principles is generally unachievable or unworkable. Thus, the model predictive controller 414 uses a model identified through a system identification process facilitated by the training data generator 408, the training data database 410, and the model identifier 412, described in detail below.

System identification, as facilitated by the training data generator 408, the training data database 410, and the model identifier 412, is a process of constructing mathematical models of dynamic systems. System identification provides a suitable alternative to first-principles-derived model when first principles models are unavailable or too complex for on-line MPC computations. System identification captures the important and relevant system dynamics based on actual input/output data (training data) of the system, in particular by determining model parameters particular to a building or zone to tune the model to the behavior of the building/zone. As described in detail below, the training data generator 408, the training data database 410, and the model identifier 412 each contribute to system identification by the controller 212.

The training data generator 408 is configured to generate training data by providing an excitation signal to the system. In some embodiments, the training data generator 408 provides various {dot over (Q)}_(HVAC) values to the equipment controller 416 for a number N of time steps k, and receives the measured output response of the indoor air temperature T_(ia) at each time step k from the air temperature sensor 214. The various {dot over (Q)}_(HVAC) values may be chosen by the training data generator 408 to explore the system dynamics as much as possible (e.g., across a full range of possible {dot over (Q)}_(HVAC) values, different patterns of {dot over (Q)}_(HVAC) values, etc.). In some embodiments, the training data generator 408 provides various T_(sp) values to the equipment controller 416 instead of the various values of {dot over (Q)}_(HVAC).

If the equipment controller 416 receives the various {dot over (Q)}_(HVAC) values, various control inputs T_(sp) can be generated in response. The temperature setpoint T_(sp) for each time step k is provided to the HVAC equipment 210, which operates accordingly to heat or cool the zone 200 (i.e., to influence T_(ia)). In some embodiments, the temperature setpoints T_(sp) are used by the training data generator 408 to be included in the training data. The training data generator receives an updated measurement of the indoor air temperature T_(ia) for each time step k and may also receive the outdoor air temperature T_(oa) for each time step k. The training data generator 408 thereby causes the states, inputs, and outputs of the system to vary across the time steps k and generates data corresponding to the inputs and outputs.

The inputs and outputs generated by the training data generator 408 are provided to the training data database 410. More particularly, in the nomenclature of the model of Eq. E and Eq. F above, the training data generator 408 provides inputs T_(sp) and T_(oa) and outputs {dot over (Q)}_(HVAC) and Ea for each time step k to the training data database 410.

The training data database 410 stores the inputs and outputs for each time step k provided by the training data generator 408. Each input and output is tagged with a time step identifier, so that data for the same time step can be associated together. The training data database 410 thereby collects and stores input and output data for each time step k, k=0, . . . N, or, more specifically, T_(sp)(k), T_(oa)(k), T_(ia)(k), and {dot over (Q)}_(HVAC)(k), for k, k=0, . . . , N. This data is grouped together in the training data database 410 in a set of training data Z^(N). In the notation of Eq. G and Eq. H, Z^(N)=[y(1), u(1), y(2), u(2), . . . , y(N), u(N)].

In some embodiments, the training data is refined using a saturation detection and removal process. System and methods for saturation detection and removal suitable for use to refine the training data Z^(N) are described in U.S. patent application Ser. 15/900,459, filed Feb. 20, 2018, incorporated by reference herein in its entirety. For example, as described in detail therein, the training data may be filtered by determining whether the operating capacity is in a non-transient region for a threshold amount of a time period upon determining that an error for the building zone exists for the time period, and in response to a determination that the operating capacity is in the non-transient region for at least the threshold amount of the time period, indicating the time period as a saturation period. Data from the saturation period can then be removed from the training data.

The model identifier 412 accesses the training data database 410 to retrieve the training data Z^(N) and uses the training data Z^(N) to identify a model of the system. The model identifier 412 includes a system parameter identifier 418 and a gain parameter identifier 420. As shown in detail in FIG. 5 and discussed in detail with reference thereto, the system parameter identifier 418 carries out a first step of system identification, namely identifying the model parameters, while the gain parameter identifier 420 carries out the second step, namely determining a Kalman gain estimator. The model parameters and the Kalman gain estimator are included in an identified model of the system, and that model is provided to the model predictive controller 414. The model predictive controller can thus facilitate the control of the HVAC equipment 210 as described above.

Referring now to FIG. 5, a detailed view of the model identifier 412 is shown, according to an exemplary embodiment. As mentioned above, the model identifier 412 includes the system parameter identifier 418 and the gain parameter identifier 420. The system parameter identifier 418 is structured to identify the matrices A, B, C, D of Eqs. G and H, i.e., the values of θ={θ₁, θ₂, θ₃, θ₄, θ₅, θ₆}. In the embodiment described herein, this corresponds to finding the values of C_(ia), C_(m), R_(mi), R_(oi), K_(p,j), and K_(i,j).

The system parameter identifier 418 includes a model framework identifier 422, a prediction error function generator 424, and an optimizer 426. The model framework identifier 422 identifies that the model of the system, denoted as M(θ), corresponds to the form described above in Eqs. G and H, i.e.,

{dot over (x)}(t)=A _(c)(θ)x(t)+B _(c)(θ)u(t);   (Eq. G)

y(t)=C _(c)(θ)x(t)+D _(c)(θ)u(t);   (Eq. H).

The model framework identifier 422 thereby determines that the system parameter identifier 418 has the goal of determining a parameter vector {circumflex over (θ)}_(N) from the set of θ∈

⊂

^(d), where

is the set of admissible model parameter values. The resulting possible models are given by the set: M={

(θ), θ∈

}. The goal of the system parameter identifier 418 is to select a parameter vector {circumflex over (θ)}_(N) from among possible values of 0 that best matches the model to the physical system (i.e., the vector θ is a list of variables and the vector {circumflex over (θ)}_(N) is a list of values), thereby defining matrices A, B, C, and D. The model framework identifier 422 also receives training data Z^(N) and sorts the training data (i.e., T_(sp)(k), T_(oa)(k), T_(ia)(k), and {dot over (Q)}_(HVAC)(k), for k, k=0, . . . , N) into the notation of Eq. G-H as input/output data Z^(N)=[y(1), u(1), y(2), u(2), . . . , y(N), u(N)].

The prediction error function generator 424 receives the model framework M={

(θ), θ∈

} and the training data Z^(N) from the model framework identifier 422. The prediction error function generator 424 applies a prediction error method to determine the optimal parameter vector {circumflex over (θ)}_(N). In general, prediction error methods determine the optimal parameter vector {circumflex over (θ)}_(N) by minimizing some prediction performance function V_(N)(θ,Z^(N)) that is based in some way on the difference between predicted outputs and the observed/measured outputs included in the training data Z^(N). That is, the parameter estimation θ_(N) is determined as:

{circumflex over (θ)}_(N)={circumflex over (θ)}_(N)(Z ^(N))=arg

V _(N)(θ, Z ^(N)).

The prediction error function generator 424 use one or more of several possible prediction error approaches to generate a prediction performance function V_(N)(θ,Z^(N)). In the embodiment shown, the prediction error function generator applies a simulation approach. In the simulation approach, the prediction error function generator 424 uses the model

(θ), the input trajectory [u(1),u(2), . . . ,u(N)], and an initial state x(0) to produce predicted outputs in terms of 0. That is, the prediction error function generator 424 predicts:

[ŷ(1|0,θ),ŷ(2|0,θ) . . . ŷ(k|0,θ) . . . , ŷ(N|0, θ)],

where ŷ(k|0, θ) denotes the predicted output at time step k given the training data from time 0 and the model

(θ). The prediction error function generator 424 then calculates a prediction error at each time step k is given by ε(k, θ):=y(k)−ŷ(k|0, θ). The prediction error function generator 424 then squares the two-norm of each prediction error ε(k θ0) and sums the results to determine the prediction performance function, which can be written as:

V _(N)(θ, Z ^(N))=Σ_(k=1) ^(N) ∥y(k)−ŷ(k|0, θ)∥₂ ²   (Eq. I).

In an alternative embodiment, the prediction error function generator 424 applies a one-step-ahead prediction error method to generate the prediction performance function V_(N)(θ, Z^(N)). In the one-step-ahead prediction error method, the prediction error function generator 424 uses past input-output data and the model

(θ) the model to predict the output one step ahead in terms of θ. That is, in the one-step ahead prediction error method, the prediction error function generator 424 generates one-step ahead predictions ŷ(k|k−1, θ), which denotes the predicted output at time step k given the past input-output sequence Z^(k−1) and using parameters θ. The one-step ahead prediction ŷ(k|k−1, θ) is then compared to the measured output y(k) by the prediction error function generator 424 to determine the prediction error at k, defined as ε(k,θ):=y(k)−ŷ(k|k−1, θ). The prediction error function generator 424 then squares the two-norm of the prediction errors for each k and sums the results, generating a prediction performance function that can be expressed in a condensed form as:

$\begin{matrix} {{V_{N}\left( {\theta,\ Z^{N}} \right)} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}{{{{y(k)} - {\overset{\hat{}}{y}\left( {\left. k \middle| {k - 1} \right.,\ \theta} \right)}}}_{2}^{2}.}}}} & \left( {{Eq}.\mspace{11mu} J} \right) \end{matrix}$

In other alternative embodiments, the prediction error function generator 424 uses a multi-step ahead prediction error approach to generate the prediction performance function. The multi-step ahead prediction error approach is described in detail below with reference to the gain parameter identifier 420 and FIGS. 7-8.

The prediction error function generator 424 then provides the performance function V_(N)(θ,Z^(N)) (i.e., from Eq. I or Eq. Jin various embodiments) to the optimizer 426.

The optimizer 426 receives the prediction error function generated by the prediction error function generator 424 and optimizes the prediction error function in θ to determine {circumflex over (θ)}_(N). More specifically, the optimizer 426 finds the minimum value of the prediction error function V_(N)(θ,Z^(N)) as θ is varied throughout the allowable values of θ∈

. That is, the optimizer 426 determines {circumflex over (θ)}_(N) based on:

{circumflex over (θ)}_(N)={circumflex over (θ)}_(N)(Z ^(N))=arg

V_(N)(θ, Z ^(N)).

The optimizer 426 then uses {circumflex over (θ)}_(N) to calculate the matrices A, B, C, and D. The system parameter identifier 418 then provides the identified matrices A, B, C, D to the gain parameter identifier 420.

The gain parameter identifier 420 receives the model with the matrices A, B, C, D (i.e., the model parameters) from system parameter identifier 418, as well as the training data Z^(N) from the training data database 410, and uses that information to identify the gain parameters. The gain parameter identifier 420 includes an estimator creator 428, a prediction error function generator 430, and an optimizer 432.

The estimator creator 428 adds a disturbance model and introduces a Kalman estimator gain to account for thermal dynamics of the system, for example for the influence of {dot over (Q)}_(other) on the system. The estimator creator 428 generates an augmented model with disturbance state d, given by:

${\begin{bmatrix} {\overset{.}{x}(t)} \\ {\overset{.}{d}(t)} \end{bmatrix} = {{\begin{bmatrix} A_{c} & B_{d} \\ 0 & 0 \end{bmatrix}\begin{bmatrix} {x(t)} \\ {d(t)} \end{bmatrix}} + {\begin{bmatrix} B_{c} \\ 0 \end{bmatrix}{u(t)}}}};$ ${y(t)} = {{\left\lbrack {C_{c}\ C_{d}} \right\rbrack \begin{Bmatrix} {x(t)} \\ {d(t)} \end{Bmatrix}} + {D_{c}{u(t)}}}$

where the parameters A_(c), B_(c), C_(c), and D_(c) are the matrices A, B, C, D received from the system parameter identifier 418 and the disturbance model is selected with

$B_{d} = \frac{1}{c_{ia}}$

and C_(d)=0.

The estimator creator 428 then converts the model to a discrete time model, for example using 5-minute sampling periods, resulting in the matrices A_(di)s, B _(dis), C_(dis), D_(dis) and the disturbance model discrete time matrix B_(d) _(dis) . The estimator creator 428 then adds a parameterized estimator gain, resulting in the following model:

$\begin{matrix} {{\begin{bmatrix} {\overset{\hat{}}{x}\left( {t + 1} \middle| t \right)} \\ {\overset{\hat{}}{d}\left( {t + 1} \middle| t \right)} \end{bmatrix} = {{\begin{bmatrix} A_{dis} & B_{d_{dis}} \\ 0 & I \end{bmatrix}\begin{bmatrix} {\overset{\hat{}}{x}\left( t \middle| {t - 1} \right)} \\ {\overset{\hat{}}{d}\left( t \middle| {t - 1} \right)} \end{bmatrix}} + {\begin{bmatrix} B_{dis} \\ 0 \end{bmatrix}{u(t)}} + {\underset{\underset{= {:{K{(\varphi)}}}}{}}{\begin{bmatrix} {K_{x}(\varphi)} \\ {K_{d}(\varphi)} \end{bmatrix}}\left( {{y(t)} - {\overset{\hat{}}{y}\left( t \middle| {t - 1} \right)}} \right)}}};} & \left( {{Eq}.\mspace{11mu} K} \right) \\ {\mspace{85mu} {{\overset{\hat{}}{y}\left( t \middle| {t - 1} \right)} = {{\left\lbrack {C_{dis}\ 0} \right\rbrack \begin{bmatrix} {\overset{\hat{}}{x}\left( t \middle| {t - 1} \right)} \\ {\overset{\hat{}}{d}\left( t \middle| {t - 1} \right)} \end{bmatrix}} + {D_{dis}{{u(t)}.}}}}} & \left( {{Eq}.\mspace{11mu} L} \right) \end{matrix}$

The matrix K(ϕ) is the estimator gain parameterized with the parameter vector ϕ where:

${{K_{x}(\varphi)} = \begin{bmatrix} \varphi_{1} & \varphi_{2} \\ \varphi_{3} & \varphi_{4} \\ \varphi_{5} & \varphi_{6} \end{bmatrix}};$ K_(d)(φ) = [φ₇ φ₈].

In this notation, {circumflex over (x)}(t+1|t) is an estimate of the state at time t+1 obtained using the Kalman filter and made utilizing information at sampling time t. For example, with a sampling time of five minutes, {circumflex over (x)}(t+1|t) is an estimate of the state five minutes after the collection of the data that the estimate is based on. The goal of the gain parameter identifier is to identify parameters {circumflex over (ϕ)}_(N) (i.e., a vector of for each of ϕ₁ . . . ϕ₈) that make the model best match the physical system.

The estimator creator 428 then provides the discrete time model with estimator gain (i.e., Eqs. K-L) to the prediction error function generator 430. The prediction error function generator receives the model from the estimator creator 428 as well as the training data Z^(N) from the training data database 410, and uses the model (with the estimator gain) and the training data Z^(N) to generate a prediction performance function.

The prediction error function generator 430 follows a multi-step ahead prediction error method to generate a predication performance function V_(N)(ϕ, Z^(N)). The multi-step ahead prediction error method is illustrated in FIGS. 7-8 and described in detail with reference thereto. As an overview, in the multi-step-ahead prediction error method, the prediction error function generator 430 uses past input-output data and the model

(θ) the model to predict the output multiple step ahead in terms of ϕ. That is, in the multi-step ahead prediction error method, the prediction error function generator 430 generates multi-step ahead predictions ŷ(k 30 h|k −1, ϕ), which denotes the predicted output at time step k+h given the past input-output sequence Z^(k−1) and using parameters ϕ. The index h corresponds the number of steps ahead the prediction is made, and for each time step k predictions are made for h=0, . . . , h_(max) (i.e., when h=2, the prediction is three steps ahead because h is indexed from zero).

Each multiple multi-step ahead prediction ŷ(k+h|k 1, ϕ) is then compared to the corresponding measured output y(k) by the prediction error function generator 430 to determine the prediction error at k, defined as ε(k, θ):=y(k)−ŷ(k+h|k−1, ϕ). The prediction error function generator 430 then squares the two-norm of the prediction errors for each k and sums the results, in some embodiments using an weighting function w(h). The prediction error function generator 430 thereby generates a prediction performance function that can be expressed in a condensed form as:

$\begin{matrix} {{V_{N}\left( {\varphi,Z^{N}} \right)} = {\sum\limits_{k = 1}^{N - h_{\max} + 1}{\sum\limits_{h = 0}^{h_{\max}}{{w(h)}{{{y\left( {k + h} \right)} - {{\overset{\hat{}}{y}\left( {k + {h\left. {{k - 1},\varphi} \right)}} \right.}_{2}^{2}.}}}}}}} & \left( {{Eq}.\mspace{14mu} M} \right) \end{matrix}$

The multi-step ahead prediction error method is described in more detail below with reference to FIGS. 7-8. In alternative embodiments, the prediction error function generator 430 follows the simulation approach or the one-step ahead prediction error approach discussed above with reference to the prediction error function generator 424.

The prediction error function generator 430 then provides the prediction performance function (i.e., Eq. M) to the optimizer 432. The optimizer 432 receives the prediction error function V_(N)(ϕ, Z^(N)) generated by the prediction error function generator 430 and optimizes the prediction error function in ϕ to determine {circumflex over (ϕ)}_(N). More specifically, the optimizer 426 finds the minimum value of the prediction error function V_(N)(ϕ, Z^(N)) as ϕ is varied throughout the allowable values of ϕ. In some cases, all real values of ϕ are allowable. That is, the optimizer 426 determines {circumflex over (ϕ)}_(N) based on:

{circumflex over (θ)}_(N)={circumflex over (ϕ)}_(N)(Z ^(N))=arg min_(ϕ) V _(N)(ϕ,Z ^(N)).

The optimizer 432 then uses {circumflex over (ϕ)}_(N) to calculate the matrices K_(x)(ϕ) and K_(d)(ϕ), resulting in a fully identified model. The gain parameter identifier 420 provides the identified model to the model predictive controller 414.

In some embodiments, the prediction error function generator 430 reconfigures the multi-step ahead prediction problem by defining augmented vectors that allow the multi-step ahead prediction performance function (Eq. M) to be recast in an identical structure to the single-step ahead prediction performance function (Eq. d). Existing software toolboxes and programs (e.g., Matlab system identification toolbox) configured to handle the single-step ahead prediction error approach can then be used to carry out the multi-step ahead prediction error approach. To reconfigure the problem for that purpose, the prediction error function generator 430 considers, the system model of the form:

x(k+1)=Ax(k)+Bu(k);

y(k)=Cx(k)+Du(k).

where the one-step prediction of {circumflex over (x)}(k+1|k) using a steady-state Kalman gain is:

{circumflex over (x)}(k+1|k)=A{circumflex over (x)}(k|k−1)+Bu(k)+K(y(k)−C{circumflex over (x)}(k|k−1)−Du(k));

ŷ(k|k−1)=C{circumflex over (x)}(k|k−1)+Du(k).

In the multi-step prediction Kalman gain system identification problem, the complete pattern of the algebraic manipulations is shown by the 4-step prediction. The prediction error function generator 430 considers a case with four input data points and four output data-points starting from time h=0 to time h=3, so that h_(max)=3. The one-step prediction (with the prediction error function generator 430 given x0) is given by the equation:

{circumflex over (x)}(1|0)=Ax0+Bu(0)+K(y(0)−Cx0−Du(0));

ŷ(0|0)=Cx0+Du(0).

The prediction of the second step is

{circumflex over (x)}(2|0)=A{circumflex over (x)}(1|0)+Bu(1)=A(Ax0+Bu(0)+K(y(0)−Cx0 −Du(0)))+Bu(1);

ŷ(1|0)=C{circumflex over (x)}(1|0)+Du(1)=C(Ax0+Bu(0)+K(y(0)−Cx0−Du(0)))+Du(1).

The prediction of the third step is

${{\overset{\hat{}}{x}\left( 3 \middle| 0 \right)} = {{{A{\overset{\hat{}}{x}\left( 2 \middle| 0 \right)}} + {B{u(2)}}} = {{A\left( {{A\left( {{Ax0} + {B{u(0)}} + {K\left( {{y(0)} - {Cx0} - {D{u(0)}}} \right)}} \right)} + {B{u(1)}}} \right)} + {{Bu}(2)}}}};$ ${\hat{y}\left( 2 \middle| 0 \right)} = {{{C{\overset{\hat{}}{x}\left( 2 \middle| 0 \right)}} + {{Du}(2)}} = {{C\left( {{A\left( {{Ax0} + {B{u(0)}} + {K\left( {{y(0)} - {Cx0} - {D{u(0)}}} \right)}} \right)} + {B{u(1)}}} \right)} + {{{Du}(2)}.}}}$

The forth step prediction is

${{\overset{\hat{}}{x}\left( 4 \middle| 0 \right)} = {{{A{\overset{\hat{}}{x}\left( 3 \middle| 0 \right)}} + {{Bu}(3)}} = {{A\left( {{A\left( {{A\left( {{{Ax}\; 0} + {B{u(0)}} + {K\left( {{y(0)} - {Cx0} - {D{u(0)}}} \right)}} \right)} + {B{u(1)}}} \right)} + {{Bu}(2)}} \right)} + {{Bu}(3)}}}};$ ${\hat{y}\left( 3 \middle| 0 \right)} = {{{C{\overset{\hat{}}{x}\left( 3 \middle| 0 \right)}} + {{Du}(3)}} = {{C\left( {{A\left( {{A\left( {{{Ax}\; 0} + {B{u(0)}} + {K\left( {{y(0)} - {Cx0} - {D{u(0)}}} \right)}} \right)} + {B{u(1)}}} \right)} + {{Bu}(2)}} \right)} + {{{Du}(3)}.}}}$

With these 4-step predictions, the pattern needed to cast the multi-step prediction problem as a 1-step prediction is revealed. By aggregating the matrices multiplying x0, y(0), u(0), u(1), u(2), and u(3), the pattern revealed is:

{circumflex over (x)}(1|0)=Ax0+Bu(0)+K(y(0)−Cx0−Du(0));

{circumflex over (x)}(2|0)=(A ² −AKC)x0+(AB−AKD)u(0)+Bu(1)+AKy(0);

{circumflex over (x)}(3|0)=(A ³ −A ² KC)x0+(A ² B−A ² KD)u(0)+ABu(1)+Bu(2)+A ² Ky(0);

{circumflex over (x)}(4|0)=(A ⁴ −A ³ KC)x0+(A ³ B−A ³ KD)u(0)+A ² Bu(1)ABu(2)+Bu(3)+A ³ Ky(0);

ŷ(0)=Cx0+Du(0);

ŷ(1|0)=(CA−CKC)x0+(CB−CKD)u(0)+Du(1)+CKy(0);

ŷ(2|0)=(CA ² −CAKC)x0+(CAB−CAKD)u(0)+CBu(1)+Du(2)+CAKy(0);

ŷ(3|0)=(CA ³ −CA ² KC)x0+(CA ² B−CA ² KD)u(0)+CABu(1)+CBu(2)+Du(3)+CA ² Ky(0).

Based on that pattern, the prediction error function generator 430 defines the following vectors:

${\overset{\sim}{u} = \begin{bmatrix} {u(0)} \\ {u(1)} \\ {u(2)} \\ {u(3)} \\ {y(0)} \end{bmatrix}},{{\overset{\overset{\sim}{\hat{}}}{y}(0)} = \begin{bmatrix} {\overset{\hat{}}{y}(0)} \\ {\overset{\hat{}}{y}\left( 1 \middle| 0 \right)} \\ {\overset{\hat{}}{y}\left( 2 \middle| 0 \right)} \\ {\overset{\hat{}}{y}\left( 3 \middle| 0 \right)} \end{bmatrix}},{{\overset{˜}{y}(0)} = \begin{bmatrix} {y(0)} \\ {y(1)} \\ {y(2)} \\ {y(3)} \end{bmatrix}},{{\overset{\hat{}}{x}\left( 1 \middle| 0 \right)}\mspace{14mu} {and}\mspace{14mu} x\; 0\mspace{14mu} {remain}\mspace{14mu} {{unchanged}.}}$

The new system that has the 4-step prediction casted into a one-step prediction which can be analyzed by the prediction error function generator 430 using an existing system identification software product as:

${\overset{\hat{}}{x}\left( 1 \middle| 0 \right)} = {{Ax0} + {\begin{bmatrix} B & 0 & 0 & 0 & 0 \end{bmatrix}{\overset{\sim}{u}(0)}} + \left\lbrack {\begin{matrix} K & 0 & 0 & \left. 0 \right\rbrack \end{matrix}\left( {{{\overset{\hat{}}{y}(0)} - {\overset{\overset{\sim}{\hat{}}}{y}(0)}};{{\overset{\overset{\sim}{\hat{}}}{y}(0)} = {\begin{bmatrix} C \\ \left( {{CA} - {CKC}} \right) \\ \left( {{CA^{2}} - {CAKC}} \right) \\ \left( {{CA^{3}} - {CA^{2}KC}} \right) \end{bmatrix}\mspace{506mu} \mspace{191mu} x{\quad{0 + {\left\lbrack \begin{matrix} D & 0 & 0 & 0 & 0 \\ \left( {{CB} - {CKD}} \right) & D & 0 & 0 & {CK} \\ \left( {{C{AB}} - {CAKD}} \right) & {CB} & D & 0 & {CAK} \\ \left( {{CA^{2}B} - {CA^{2}{KD}}} \right) & {CAB} & {CB} & D & {CA^{2}K} \end{matrix} \right\rbrack {{\overset{\overset{\sim}{\hat{}}}{y}(0)}.}}}\mspace{14mu}}}}} \right.} \right.}$

In order to have the general formulation at time k for predicting h_(max) step ahead in time, this four-step example can be extrapolated to define the general augmented input and output vectors as:

${{\overset{\overset{\sim}{\hat{}}}{y}(k)} = {{\begin{bmatrix} C \\ (CA - CKC) \\ \left( CA^{2} - CAKC \right) \\ \vdots \\ \left( CA^{h_{\max}} - CA^{h_{\max} - 1}KC \right) \end{bmatrix}{\overset{\hat{}}{x}\left( k \middle| {k - 1} \right)}} + \mspace{34mu} {\left\lbrack \begin{matrix} C & 0 & 0 & 0 & 0 & 0 & 0 \\ \left( {{CA} - {CKD}} \right) & D & 0 & 0 & 0 & 0 & {CK} \\ \left( {{CAB} - {CAKD}} \right) & {CB} & D & 0 & 0 & 0 & {CAK} \\ \left( {{{CA}^{2}B} -} \right. & {CAB} & \ddots & \ddots & 0 & 0 & {{CA}^{2}K} \\ \left. {{CA}^{2}{KD}} \right) & \; & \; & \; & \; & \; & \; \\ \vdots & \vdots & \ddots & {CB} & D & 0 & \vdots \\ \left( {{CA^{h_{\max} - 1}B} -} \right. & {{CA}^{h_{\max} - 2}B} & \ldots & {CAB} & {CB} & D & {{CA}^{h_{\max} - 1}K} \\ \left. {{CA}^{h_{\max} - 1}{KD}} \right) & \; & \; & \; & \; & \; & \; \end{matrix} \right\rbrack {\overset{\sim}{u}(k)}}}};$ ${{\overset{\sim}{u}(k)} = \begin{bmatrix} {u(k)} \\ {u(k + 1)} \\ \vdots \\ {u\left( k + h_{\max} \right)} \\ {y(k)} \end{bmatrix}},{{\overset{\overset{\sim}{\hat{}}}{y}(k)} = \begin{bmatrix} {\overset{\hat{}}{y}\left( k \middle| k - 1 \right)} \\ {\overset{\hat{}}{y}\left( k + 1 \middle| k - 1 \right)} \\ \vdots \\ {\overset{\hat{}}{y}\left( k + h_{\max} \middle| k - 1 \right)} \end{bmatrix}},{{\overset{\sim}{y}(k)} = \begin{bmatrix} {y(k)} \\ {y\left( {k + 1} \right)} \\ \vdots \\ {y\left( {k + h_{\max}} \right)} \end{bmatrix}}$

With these definition, the general formulation at time k for predicting h_(max) steps ahead in time is:

{circumflex over (x)}(k+1|k)=A{circumflex over (x)}(k|k−1)+[B0]ũ(k)+[K0 . . . 0]({tilde over (y)}(k)−{circumflex over ({tilde over (y)})}(k).

As described above, in the multi-step ahead prediction error method the prediction error function generator 430 generates a function of the form:

$\begin{matrix} {{V_{N}\left( {\varphi,Z^{N}} \right)} = {\sum\limits_{k = 1}^{N - h_{\max} + 1}{\sum\limits_{h = 0}^{h_{\max}}{{w(h)}{{{y\left( {k + h} \right)} - {{\overset{\hat{}}{y}\left( {k + {h\left. {{k - 1},\varphi} \right)}} \right.}_{2}^{2}.}}}}}}} & \left( {{Eq}.\mspace{14mu} M} \right) \end{matrix}$

If w(h)≡1 for all h, and using the augmented input and output vectors defined above, the multi-step ahead prediction performance function can be reconfigured into the following one-step ahead prediction performance function by the prediction error function generator 430:

${V_{N}\left( {\theta,Z^{N}} \right)} = {{V_{N}\left( {\theta,Z^{N}} \right)} = {\sum\limits_{k = 1}^{N - h_{\max} + 1}{{{\overset{\sim}{y}(k)} - {\overset{\overset{\sim}{\hat{}}}{y}\left( {k,\theta} \right)}}}_{2}^{2}}}$

The prediction error function generator 430 then uses this reconfigured format of the prediction performance function with existing software toolboxes suited for the one-step ahead prediction error approach. The prediction error function generator 430 may include machine-readable media storing computer code executable to apply such software.

System Identification Methods

Referring now to FIG. 6, a flowchart of a process 600 for system identification is shown, according to an exemplary embodiment. The process 600 can be carried out by the controller 212 of FIGS. 2 and 4.

At step 602, the controller 212 applies an excitation signal to the HVAC equipment 210. For example, the training data generator 408 may vary the {dot over (Q)}_(HVAC) values supplied to the equipment controller 416, causing an excitation signal to be generated in the temperature setpoint T_(sp) inputs provided to the HVAC equipment 210. In general, the excitation signal is designed to test the system in a way to provide robust data for use in system identification.

At step 604, training data is collected and stored by the controller 212. Training data includes measureable temperature readings, i.e., T_(oa) and T_(ia), controller-determined values {dot over (Q)}_(HVAC) and T_(sp) for each of a plurality of time steps k, k=0, . . . , N. The training data therefore includes inputs u(k) and the outputs y(k) for the time period. The training data is received from temperature sensors 214, 216, training data generator 408, and/or equipment controller 416 and stored in training data database 410.

At step 606, the controller 212 identifies the model parameters for the system. That is, as discussed in detail above, the controller 212 determines the matrices A(θ), B(θ), C(θ), and D(θ) that minimize a prediction performance function V_(N)(Z^(N), θ) for the model:

{dot over (x)}+(t)=A _(c)(θ)x(t)+B _(c)(θ)u(t);   (Eq. G)

y(t)=C _(c)(θ)x(t)+D _(c)(θ)u(t);   (Eq. H).

In identifying the model parameters, a simulation approach or a one-step-ahead prediction error approach is followed in some embodiments. These approaches are described in detail above with reference to the prediction error function generator 424 of FIG. 5. In other embodiments, the model parameters are determined at step 606 using a multi-step ahead prediction error method, described in detail with reference to FIGS. 7-8.

At step 608, the controller 212 identifies the gain estimator parameters. That is, the controller 212 determines the matrices K_(x) and K_(d) of Eq. K above. In preferred embodiments, the controller 212 uses the multi-step ahead prediction error method to find the matrices K_(x) and K_(d). The multi-step ahead prediction error method is described in detail below with reference to FIGS. 7-8. In alternative embodiments, a simulation approach or a one-step-ahead prediction error approach is followed to find the matrices K_(x) and K_(d).

At step 610, the identified model is validated by the controller 212. The controller 212 uses the identified model to generate control signal inputs T_(sp) for the HVAC equipment 210 using model predictive control. The controller then monitors the temperature measurements T_(oa) and T_(ia) from temperature sensors 214, 216, the input T_(sp), and the value {dot over (Q)}_(HVAC) to determine how well the model matches system behavior in normal operation. For example, the training data database 410 may collect and store an addition set of training data that can be used by the model identifier 412 to validate the model. If some discrepancy is determined, the identified model may be updated. The identified model can thereby by dynamically adjusted to adjust for changes in the physical system.

Referring now to FIGS. 7-8 the multi-step ahead prediction error approach for use in system identification is illustrated, according to an exemplary embodiment. In FIG. 7, a flowchart of a process 700 for identifying system parameters using the multi-step ahead prediction error approach is shown, according to an exemplary embodiment. FIG. 8 shows an example visualization useful in explaining process 700. Process 700 can be carried out by the system parameter identifier 418 and/or the gain parameter identifier 420 of FIG. 5. In the embodiment described herein, the process 700 is implemented with the gain parameter identifier 420.

Process 700 begins at step 702, where the gain parameter identifier 420 receives training data Z^(N)=[y(1), u(1), y(2), u(2), . . . , y(N), u(N)] from the training data database 410. The training data includes measured outputs y(k) (i.e., T_(ia)(k) and {dot over (Q)}_(HVAC)(k)) and inputs u(k) (i.e., T_(oa)(k) and T_(sp)(k)) for each time step k, k=1, . . . , N. N is the number of samples in the training data. The gain parameter identifier 420 also receives the system model from the system parameter identifier 418.

At step 704, the prediction error function generator 430 uses the training data for a time step k to predict outputs ŷ for each subsequent time step up to k+h_(max). The value h_(max) corresponds to the number of steps ahead the predictions are made, referred to herein as the prediction horizon. Because h_(max) is indexed from zero in Eq. M above, the prediction horizon is one more than the value of hmax. For example, in the case shown in FIG. 8 and described below, predictions are made three steps ahead, corresponding to h_(max)=2 in the notation of Eq. D and a prediction horizon of three. The prediction horizon may be any integer greater than one, for example four or eight. The prediction horizon can be tuned experimentally, to determine an ideal prediction horizon. For example, too long of a prediction horizon may lead to poor prediction while too short of a prediction horizon may suffer the same limitations as the one-step ahead prediction error method mentioned above. In some cases, a prediction horizon of eight is preferred.

More specifically, at each step 704 the predicted outputs [ŷ(k|k−1), ŷ(k+1|k−1), . . . ŷ(k+h_(max)|k−1)] are predicted based on the past training data (i.e., through step k−1), denoted as Z^(k−1), along with future inputs [u(k), u(k+1) . . . u(k+h_(max))]. These predictions are made using the model M(ϕ), such that predicted outputs ŷ depend on ϕ.

To illustrate the predictions of step 704, FIG. 8 shows a simplified visualization in which y(k) and ŷ(k) are depicted as scalar values for the sake of simplified explanation. In FIG. 8, the graph 800 plots the values of y and ŷ over time t for five time steps past a starting time t=0. The solid circles 802 represent measured outputs y(t) from the training data. The unfilled boxes 804 represent predicted outputs ŷ(t|0), that is, the outputs predicted for each time step based on the input/output data available at time t=0 (e.g., y(0)). The dashed lines represent the propagation of the predictions; for example, graph 800 includes three unfilled boxes 804 connected by a dashed line to the solid circle 802 corresponding to y(0). This shows that the predictions ŷ(t|0), 1≤t≤3, represented by the unfilled boxes 804 were based on the measured value of y(0).

At step 706, the prediction error function generator 430 compares the predicted outputs ŷ to the measured outputs y for each future step up to k+h_(max) (i.e., for all predicted outputs ŷ generated at step 704). More specifically, an error term for each step may be defined as y(k+h)−ŷ(k+h|k−1,ϕ). Because y and ŷ are vectors, the two-norm of this error term may be taken and squared to facilitate comparison between prediction errors as scalars, such that the error term becomes ∥y(k+h)−ŷ(k+h|k−1, ϕ)∥₂ ³. This term appears in Eq. M above.

As shown in FIG. 8, step 706 can be understood as measuring the distance between, for example, each unfilled box 804 and the corresponding solid circle 802 (i.e., the unfilled box 804 and the solid circle 802 at the same time t). Thus, in the example of FIG. 8, step 706 includes calculating three error terms.

At step 708, the error terms are weighted based on a weighting function w(h). The weighting function w(h) allows the prediction errors to be given more or less weight depending on how many steps ahead the prediction is. The weighting function w(h) is preferably a monotonically decreasing function of h, so that farther-out-in-time predictions have less influence on the prediction error. In some embodiments, the weighting function w(h)=1. Step 708 thereby corresponds the w(h) term in Eq. M above.

The process 700 then returns to step 704 to repeat steps 704-706 for each value of k, k=1, As illustrated in FIG. 8, repeating step 704 corresponds to generating the predictions represented by the unfilled circles 808 and the unfilled triangles 810. The unfilled circles 808 chart the predictions based on the output data available at time t=1, i.e., ŷ(t|1), for t=2, 3, 4. The unfilled triangles chart the predictions based on the output data available at time t=2, i.e., ŷ(t|2), for t=3, 4, 5. Process 700 therefore involves making multiple predictions for most time steps: for example, FIG. 8 shows three separate predictions for time t=3.

At step 706, the prediction error function generator 430 again compares the predicted outputs ŷ for the new value of k to the measured outputs y for each future step up to k+h_(max) to define the error term ∥y(k+h)−ŷ(k+h|k−1, 0)∥₂ ² as included in Eq. M. At step 708, the terms are again weighted by the weighting function w(h). The weighting function w(h) may be the same for each k.

In the notation of Eq. M, each iteration of steps 704-708 thus corresponds to steps necessary to generate the values used by the inner (right) summation indexed in h, while repetition of the steps 704-708 corresponds to the iteration through k represented in the outer (left) summation. At step 710, then, these summations are executed. In other words, the system identification circuit 108 sums the weighted error terms generated by steps 704-708 to generate a prediction performance function as:

$\begin{matrix} {{V_{N}\left( {\varphi,Z^{N}} \right)} = {\sum\limits_{k = 1}^{N - h_{\max} + 1}{\sum\limits_{h = 0}^{h_{\max}}{{w(h)}{{{y\left( {k + h} \right)} - {{\overset{\hat{}}{y}\left( {k + {h\left. {{k - 1},\varphi} \right)}} \right.}_{2}^{2}.}}}}}}} & \left( {{Eq}.\mspace{14mu} M} \right) \end{matrix}$

The prediction performance function is a function of the input data Z^(N) and the parameter variable ϕ. Typically, the input data Z^(N) is given (i.e., received by the model identifier 412 and used in the calculation of error terms as described above). Thus, the prediction performance function is primarily a function of ϕ.

At step 712, the prediction performance function V_(N)(ϕ, Z^(N)) is minimized to find an optimal parameter vector {circumflex over (θ)}_(N)=arg

V_(N)(ϕ, Z^(N)). Any minimization procedure may be followed. The result of step 712 is a vector {circumflex over (ϕ)}_(N) of identified model parameters that tune the model

({circumflex over (ϕ)}_(N)) to accurately predict system evolution multiple steps ahead. At step 714, the model identifier 412 provides the identified system model (i.e.,

({circumflex over (ϕ)}_(N))) to the model predictive controller 414 for use in generating control inputs for the HVAC equipment 210.

According to various embodiments, process 700 is run once at set-up to establish the system model, run periodically to update the system model, or run repeatedly/continuously to dynamically update the system model in real time.

Experimental Results

To illustrate the advantages of the systems and methods described above, the following experimental results are included and shown in FIGS. 9-18. The HVAC system 100 and the building 10 were put through two experiments: a heating and a cooling experiment.

Heating Experiment

In the heating experiment, a simulated HVAC system 100 is in a heating mode to heat a simulated building 10. Because the system is simulated the actual values of the system parameters and the unmeasured time-varying disturbances ({dot over (Q)}_(other)) are known in the experiment for sake of comparison to the identified model.

To start, the controller 212 provides excitation signal to the HVAC equipment 210. The excitation signal 902 illustrated on graph 900 in FIG. 9 varies the temperature setpoint T_(sp) using a pseudorandom binary signal that varies between the maximum and minimum allowable temperatures in the comfort zone (T_(max)=25° C., T_(min)=23° C.).

While the excitation signal is applied to the HVAC equipment 210, training data is collected and stored in the controller 212 for each time step k as described above. Each time step kin the heating experiment corresponds to five minutes of time (i.e., a data sample is recorded every five minutes). The training data is used by the system parameter identifier 418 to identify the model parameters as described above. In the heating experiment, the following results were found:

Identified Parameters Actual Parameters C_(ia,id) = 2.287037e+003 C_(ia) = 1.0448e+04 C_(s,id) = 3.2507187e+03 C_(s) = 3.4369e+05 R_(si,id) = 0.57426198230 R_(si) = 0.0863 R_(oi,id) = 0.69936 R_(oi) = 0.3302 τ_(I,id) = 182.74 τ_(I) = 180 K_(c,id) = 2.637 K_(c) = 12

The first step of model parameterization, carried out by the system parameter identifier 418, thereby determined the parameters to the correct order of magnitude, but some differences are present due to the time-varying disturbances (i.e., {dot over (Q)}_(other)).

Next, the Kalman gain parameters are identified by the gain parameter identifier 420. In the experiment, the gain parameters are identified using a one-step ahead prediction error method, a two-step ahead prediction error method, a five-step ahead prediction error method, a ten-step ahead prediction error method, and a fifty-step ahead prediction error method. As an example of the results, FIG. 10 shows a graph 1000 of the actual indoor temperature and the predicted indoor temperature over time as generated by the five-step ahead prediction error method. FIG. 11 shows a graph 1100 of the actual {dot over (Q)}_(HVAC) and the predicted {dot over (Q)}_(HVAC) over time as generated by the five-step ahead prediction error method. As shown in FIGS. 10 and 11, the predicted values of T_(ia) and {dot over (Q)}_(HVAC) consistently track the actual values.

The different number of steps (i.e., h_(max) values) were included to allow comparison between the parameters identified using different numbers of steps. The Kalman gains identified using the various numbers of steps are presented in the following table:

1-Step Kalman 2-Step Kalman 5-step Kalman 10-Step Kalman 50-Step Kalman T_(ia) Q_(HVAC) T_(ia) Q_(HVAC) T_(ia) Q_(HVAC) T_(ia) Q_(HVAC) T_(ia) Q_(HVAC) T_(m) 4.4287 0.3367 3.570 0.5273 3.2643 0.3119 1.1435 0.4487 −0.4660 0.1126 T_(ia) 1.3442 −0.0007 0.908 −0.0098 0.6599 −0.0128 0.4876 −0.0188 0.1696 −0.0826 I −125.5 −110.8 62.25 −105.345 73.984 −110.048 172.649 −105.768 78.550 −74.3589 d −0.0008 0.0005 −0.01 0.0003 −0.0015 0.0004 −0.0014 0.0003 −0.0003 0.0001

FIG. 12 shows a graph 1200 of the estimated building mass temperature T_(m) over time for the one-step prediction error method, the two-step prediction error method, and the fifty-step prediction error method, as well as the actual T_(m) of the simulated building 10. As the number of steps increase, the T_(m) estimates improve, following the actual T_(m) line on graph 1200 closer.

Cooling Experiment

In the cooling experiment, a simulated HVAC system 100 is in a cooling mode to cool a simulated building 10. As above, because the system is simulated the actual values of the system parameters and the unmeasured time-varying disturbances ({dot over (Q)}_(other)) are known in the experiment for sake of comparison to the identified model.

A similar procedure as the heating experiment is followed to generate models, with the Kalman gain generated using the multi-step prediction error method with a variety of number of steps (i.e., various prediction horizons h_(max)) (e.g., one step, two steps, eight steps, twelve steps, twenty steps). FIG. 13 shows output predictions generated using the one-step prediction error method compared to actual outputs, with T_(ia) graphed over time on graph 1300 and {dot over (Q)}_(HVAC) graphed over time on graph 1350. Similarly, FIG. 14 shows output predictions generated using the two-step prediction error method compared to actual outputs, with Tia graphed over time on graph 1400 and {dot over (Q)}_(HVAC) graphed over time on graph 1450.

To compare the results of the various multi-step prediction error methods (i.e., various number of steps), several metrics are used. First, a weighted mean absolute prediction error (WMAPE) metric is an exponentially weighted average of the absolute prediction error at each time step and given by:

${{{WMAPE}(k)} = \frac{\sum\limits_{i = k}^{k + N_{h} - 1}{e^{{- i}/N_{h}}{{{y(i)} - {\hat{y}\left( {ik} \right)}}}}}{\sum\limits_{i = k}^{k + N_{h} - 1}e^{{- i}/N_{h}}}},{k = 0},1,2,{\ldots \mspace{14mu}.}$

where N_(h)∈

_(>0) is the prediction horizon, y(i) is the actual output at time step i and ŷ(i|k) is the predicted output with the identified model given a measurement at time step k and the input sequence u(k), u(k+1), . . . , u(i−1). In the WMAPE equation, y is used to refer to a scalar (i.e., one of the two outputs), and the WMAPE is computed separately for both outputs. The horizon used to calculate the WMAPE in the cooling experiment was twelve. FIG. 15 shows a graph 1500 of the WMAPE for Tia for the one-step ahead prediction error method and a 12-step ahead prediction error method for comparison. FIG. 15 also shows a graph 1550 of the WMAPE for {dot over (Q)}_(HVAC) for the one-step ahead prediction error method and a 12-step ahead prediction error method for comparison.

Another metric that can be used to evaluate the results of the cooling experiment is the root mean squared prediction error (RMSPE). RMSPE is calculated for a range of values of q from zero-step ahead prediction up to N_(h)-step ahead prediction. That is, given a set of measured output values {y(0), . . . , y(M)} for M∈

_(≥0), the RMSPE is given by:

${{RMSPE}(q)} = \sqrt{\frac{\sum\limits_{i = {q + 1}}^{M}\left( {{y(i)} - {\overset{\hat{}}{y}\left( i \middle| {i - q} \right)}} \right)^{2}}{M - q}}$

for all q∈{0, . . . ,N_(h)−1}. The RMSPE helps identify the prediction error over the prediction horizon. In the example here, the RMSPE is calculated for 288 steps (i.e., N_(h)=288). FIG. 16 shows a graph 1600 of the RMSPE for Ea for the one-step ahead prediction error method and a two-step ahead prediction error method for comparison. FIG. 16 also shows a graph 1650 of the WMAPE for {dot over (Q)}_(HVAC) for the one-step ahead prediction error method and the two-step ahead prediction error method for comparison.

A third way to compare across different numbers of steps is to visualize the power of prediction. FIGS. 17 and FIG. 18 shows examples visualizations 1700 and 1800 of this third metric. To generate the visualizations 1700, 1800, ten lines of N-steps-ahead predictions are plotted using the Kalman gain generated by each multi-step ahead prediction method. That is, a first line starts x0(i.e., an initial state) and plots the N step ahead prediction, from {circumflex over (x)}(1|0) all the way to {circumflex over (x)}(N|0). The second line takes {circumflex over (x)}(1|0) and plots N steps ahead, and so on, until ten lines are plotted. The closer the lines are to being on top of each other, the better the output multi-step prediction. In the examples of FIGS. 15 and 16, the lines are plotted for twelve steps ahead (N=12).

The visualization 1700 of FIG. 17 is thereby generated for each of the one-step ahead prediction error method, the eight-step ahead prediction error method, the twelve-step ahead prediction error method, the twenty-step ahead prediction error method for the output T_(ia). The visualization 1800 of FIG. 18 is generated in the same way for each of the one-step ahead prediction error method, the eight-step ahead prediction error method, the twelve-step ahead prediction error method, the twenty-step ahead prediction error method for the output {dot over (Q)}_(HVAC).

In both visualization 1700 and visualization 1800, the eight-step prediction error method is shown to have the best results (i.e., the lines are plotted closest together), even though the lines were plotted twelve steps ahead. Thus, in some embodiments, an eight-step ahead prediction error method may be preferred (i.e., h_(max)=7). Because each time step is five minutes in the experiment, this implies that a prediction horizon of forty minutes in the Kalman gain identification is well suited for generating a model that predicts one hour (12 steps) into the future.

Heat Load Modeling Overview

Referring generally to FIGS. 19-21, methods for estimating heat disturbance Q_(other) are shown, according to some embodiments. Heat disturbance refers to heat in a building (or any space) that originates from sources beyond measurement and/or control of an environmental control system of the building. For example, heat disturbance may result from sunlight, heat radiating from electrical equipment, body heat radiation, etc. Accurately estimating heat disturbance can increase accuracy of estimations made during a model predictive control process. Without estimations of heat disturbance, a significant source of heat in a building may go unaccounted for, thus reducing accuracy of model predictive control and increasing energy usage and/or occupant discomfort.

As explained in greater detail below, heat disturbance can be modeled as a summation of a deterministic heat disturbance prediction and a stochastic heat disturbance prediction. The deterministic heat disturbance can describe a piece of a total heat disturbance that can be determined based on parameter values and initial conditions of a heat disturbance estimation problem. In some embodiments, the deterministic heat disturbance is calculated using a process for estimating deterministic load as described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, incorporated by reference herein in its entirety. However, the stochastic heat disturbance, a piece of the total heat disturbance that describes some inherent randomness of the heat disturbance, can be difficult to calculate.

In some embodiments, the deterministic heat disturbance model is obtained using a pattern recognition and linear regression strategy as described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, incorporated by reference herein in its entirety. In some embodiments, the stochastic heat disturbance model is obtained through identification of an autoregressive (AR) model separate from a system state space model used in model predictive control (MPC). In some embodiments, the stochastic heat disturbance model is obtained through identification of a model that is part of an overall state space model used in estimation and prediction during an MPC process.

Processes for Heat Load Modeling

Referring now to FIG. 19, a process 1900 for estimating historical heat disturbance is shown, according to some embodiments. By estimating historical heat disturbance based on prior data, a model can be trained to predict current and future heat disturbances for use in performing model predictive control. The historical heat disturbance can be used to train a model for determining how heat disturbance affects a space (e.g., a zone in a building). In some embodiments, some and/or all steps of process 1900 are performed by controller 212 described with reference to FIG. 2.

Process 1900 is shown to include receiving training data for a building system (step 1902), according to some embodiments. The training data for the building system may describe measurements of various environmental information taken at previous time steps and control inputs of the system, for example as described above with reference to training data generator 408 and training data database 410. For example, the measurements may include information such as building temperature measurements, building humidity measurements, occupancy measurements, electrical usage measurements, time of day, day of week, external weather conditions, etc. As heat disturbance can originate from many sources, large amounts of training data may be necessary to properly estimate historical heat disturbance. In some embodiments, the training data received in step 1902 includes time-series data. Time-series data can further indicate how heat disturbance changes over time. For example, in the middle of a day when occupants are present in the building system, a higher heat disturbance due to thermal radiation of people may be present than in the middle of the night when fewer occupants are present. In some embodiments, the training data is refined using a saturation detection and removal process. System and methods for saturation detection and removal suitable for use to refine the training data are described in U.S. patent application Ser. No. 15/900,459, filed Feb. 20, 2018, incorporated by reference herein in its entirety. In some embodiments, step 1902 is performed by controller 212.

Process 1900 is shown to include performing system identification to identify system matrices A, B, C, and D (step 1904), according to some embodiments. Based on the system matrices A, B, C, and D, a system model determined via the system identification can be used when estimating heat disturbance. In some embodiments, the system identification performed in step 1904 to identify the system model is similar to and/or the same as the system identification process as described in U.S. patent application Ser. 16/240,028, filed Jan. 4, 2019, incorporated by reference herein in its entirety. By identifying the system matrices, a system model can be captured to be used in determining heat disturbance. In general, the system matrices describe dynamics of the system. In step 1904, the system model identified can illustrate dynamics (e.g., thermal dynamics) of the building system based on the training data. In some embodiments, step 1904 is similar to and/or the same as process 600 described with reference to FIG. 6. In some embodiments, step 1904 is performed by controller 212.

Process 1900 is shown to include augmenting the system matrices with a disturbance model having two or more dimensions (step 1906), according to some embodiments. In general, an augmented system of the system identified in step 1904 is given by the following state space representation:

$\begin{matrix} {{\begin{bmatrix} \overset{.}{x(t)} \\ \overset{.}{d(t)} \end{bmatrix} = {{\begin{bmatrix} A_{c} & B_{d} \\ 0 & A_{d} \end{bmatrix}\begin{bmatrix} {x(t)} \\ {d(t)} \end{bmatrix}} + {\begin{bmatrix} B_{c} \\ B_{dd} \end{bmatrix}\mspace{14mu} {u(t)}}}};} & \left( {{Eq}.\mspace{14mu} N} \right) \\ {{{y(t)} = {{\left\lbrack {C_{c}\mspace{20mu} C_{d}} \right\rbrack \begin{bmatrix} {x(t)} \\ {d(t)} \end{bmatrix}} + {D_{c}{u(t)}}}};} & \left( {{Eq}.\mspace{14mu} O} \right) \end{matrix}$

where A_(d), B_(d), B_(dd), and C_(d) are matrices characterizing the disturbance model and the parameters A_(c), B_(c), C_(c), and D_(c) are the matrices A, B, C, D identified in step 1904. In some embodiments, the disturbance model is a parameterized model such that the parameterized model can generate accurate models of heat disturbance over repeated usages of the model given various parameters.

In some embodiments, the disturbance model is determined based on a known higher order disturbance model. The disturbance model is described in greater detail in Eq. T and Eq. U below with reference to step 1908. If the disturbance model is determined based on a known higher order disturbance model, an augmented system of the system identified in step 1904 may have the following form:

$\begin{matrix} {{\left\lbrack \begin{matrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ \overset{.}{I} \\ {\overset{.}{d}}_{1} \\ {\overset{.}{d}}_{2} \end{matrix} \right\rbrack = {{\left\lbrack \begin{matrix} {\frac{1}{C_{ia}}\left( {{- K_{p}} - \frac{1}{R_{mi}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{mi}} & \frac{K_{I}}{C_{ia}} & \frac{1}{C_{ia}} & 0 \\ \frac{1}{C_{m}R_{mi}} & {- \frac{1}{C_{m}R_{mi}}} & 0 & 0 & 0 \\ {- 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {- w^{2}} & {{- 2}\gamma} \end{matrix} \right\rbrack\left\lbrack \begin{matrix} T_{ia} \\ T_{m} \\ I \\ d_{1} \\ d_{2} \end{matrix} \right\rbrack} + {\left\lbrack \begin{matrix} \frac{K_{p}}{C_{ia}} & \; & \frac{1}{C_{ia}R_{oi}} \\ 0 & \; & 0 \\ 1 & \; & 0 \\ \; & B_{dd} & \; \end{matrix} \right\rbrack \begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}};} & \left( {{Eq}.\mspace{14mu} P} \right) \\ {{\begin{bmatrix} T_{ia} \\ {\overset{.}{Q}}_{HVAC} \end{bmatrix} = {{\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ {- K_{p}} & 0 & K_{I} & 0 & 0 \end{bmatrix}\left\lbrack \begin{matrix} T_{ia} \\ T_{m} \\ I \\ d_{1} \\ d_{2} \end{matrix} \right\rbrack} + {\begin{bmatrix} 0 & 0 \\ K_{p} & 0 \end{bmatrix}\begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}};} & \left( {{Eq}.\mspace{14mu} Q} \right) \end{matrix}$

where d₁ is a disturbance state that can estimate (or calculate) values of heat disturbance, d₂ is a disturbance state that can estimate (or calculate) a rate of change in values of the heat disturbance, w is a frequency tuning parameter, γ is a damping tuning parameter, B_(dd) is a matrix mapping a forcing input, T_(sp) is an indoor air temperature setpoint, and T_(oa) is an outdoor air temperature, and all other variables are defined as in Eq. E and Eq. F above. In the above augmented system, d₁ includes the entirety of {dot over (Q)}_(other) such that {dot over (Q)}_(other) is not explicitly identified in the augmented system. As such, if d₁ is identified, {dot over (Q)}_(other) may inherently be identified as well. Similarly, if the disturbance model is determined based on the identified disturbance model, the system identified in step 1904 may have the following form:

$\begin{matrix} {{\left\lbrack \begin{matrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ \overset{.}{I} \\ {\overset{.}{d}}_{1} \\ {\overset{.}{d}}_{2} \end{matrix} \right\rbrack = {{\left\lbrack \begin{matrix} {\frac{1}{C_{ia}}\left( {{- K_{p}} - \frac{1}{R_{mi}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{mi}} & \frac{K_{I}}{C_{ia}} & \frac{1}{C_{ia}} & 0 \\ \frac{1}{C_{m}R_{mi}} & {- \frac{1}{C_{m}R_{mi}}} & 0 & 0 & 0 \\ {- 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \theta_{1} & \theta_{2} \\ 0 & 0 & 0 & \theta_{3} & \theta_{4} \end{matrix} \right\rbrack \mspace{284mu}\left\lbrack \begin{matrix} T_{ia} \\ T_{m} \\ I \\ d_{1} \\ d_{2} \end{matrix} \right\rbrack} + {\left\lbrack \begin{matrix} \frac{K_{p}}{C_{ia}} & \frac{1}{C_{ia}R_{oi}} \\ 0 & 0 \\ 1 & 0 \\ \theta_{5} & \theta_{6} \\ \theta_{7} & \theta_{8} \end{matrix} \right\rbrack \begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}};} & \left( {{Eq}.\mspace{14mu} R} \right) \\ {{\begin{bmatrix} T_{ia} \\ {\overset{.}{Q}}_{HVAC} \end{bmatrix} = {{\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ {- K_{p}} & 0 & K_{I} & 0 & 0 \end{bmatrix}\left\lbrack \begin{matrix} T_{ia} \\ T_{m} \\ I \\ d_{1} \\ d_{2} \end{matrix} \right\rbrack} + {\begin{bmatrix} 0 & 0 \\ K_{p} & 0 \end{bmatrix}\begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}};} & \left( {{Eq}.\mspace{14mu} S} \right) \end{matrix}$

where θ₁ through θ₈ are parameters that can be identified or set to a prospected value, and all other variables being the same as above. In general, the augmented system model can be used for determining historic heat disturbances. In some embodiments, step 1906 is performed by controller 212.

Process 1900 is shown to include identifying and/or selecting parameters of the disturbance model and identifying a Kalman gain (step 1908), according to some embodiments. In some embodiments, the parameters are determined using a known higher order disturbance model. Using the known higher order disturbance model, a rate of change in estimated historical heat disturbance values can be captured. Further, periodic heat disturbances can be represented using the known higher order disturbance model. For example, solar irradiance may follow a periodic schedule with a peak around noon each day. In some embodiments, the known higher order disturbance model is a second order disturbance model of an oscillator system having two states, d₁ and d₂, where d₁ can estimate (or calculate) values of heat disturbance and d₂ can estimate (or calculate) a rate of change in values of the heat disturbance. In general, the oscillator system may have the following form:

$\begin{matrix} {{\begin{bmatrix} \overset{.}{d_{1}(t)} \\ \overset{.}{d_{2}(t)} \end{bmatrix} = {{{A_{d}\begin{bmatrix} {d_{1}(t)} \\ {d_{2}(t)} \end{bmatrix}} + {B_{dd}{u(t)}}} = {{\begin{bmatrix} 0 & 1 \\ {- w^{2}} & {{- 2}\gamma} \end{bmatrix}\begin{bmatrix} {d_{1}(t)} \\ {d_{2}(t)} \end{bmatrix}} + {B_{dd}\begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}}};} & \left( {{Eq}.\mspace{14mu} T} \right) \end{matrix}$

or compactly as:

$\begin{matrix} {{\begin{bmatrix} {\overset{\cdot}{d}}_{1} \\ {\overset{\cdot}{d}}_{2} \end{bmatrix} = {{{A_{d}\begin{bmatrix} d_{1} \\ d_{2} \end{bmatrix}} + {B_{dd}u}} = {{\begin{bmatrix} 0 & 1 \\ {- w^{2}} & {{- 2}\gamma} \end{bmatrix}\begin{bmatrix} d_{1} \\ d_{2} \end{bmatrix}} + {B_{dd}\begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}}};} & \left( {{Eq}.\mspace{11mu} T} \right) \end{matrix}$

where w is a frequency tuning parameter, γ is a damping tuning parameter, B_(dd) is a matrix mapping a forcing input, T_(sp) is an indoor air temperature setpoint, and T_(oa) is an outdoor air temperature. For example, the values of w and y can be selected to provide a user-selected period or frequency for the oscillator system, for example a period of one day that reflects oscillations in solar irradiance as described above. As such, the tuning parameters can be set to

${w = \frac{2\pi}{24 \times 3600}},{\gamma = 0},$

and the matrix that maps the forcing input B_(dd) can be set w zeroes

$\left( {{i.e.},{B_{dd} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}}} \right).$

In this example, a pure oscillator system with zero damping and a frequency corresponding to a one-day period is achieved.

In some embodiments, the disturbance model is determined via a system identification using the training data received in step 1902. Similar to using the known higher order disturbance model, two disturbance states, d₁ and d₂, represent values of heat disturbance and a rate of change in the values of the heat disturbance respectively. In general, the disturbance model determined via system identification has the following form:

$\begin{matrix} {{\begin{bmatrix} \overset{.}{d_{1}(t)} \\ \overset{.}{d_{2}(t)} \end{bmatrix} = {{{A_{d}\begin{bmatrix} {d_{1}(t)} \\ {d_{2}(t)} \end{bmatrix}} + {B_{dd}{u(t)}}} = {{\begin{bmatrix} \theta_{1} & \theta_{2} \\ \theta_{3} & \theta_{4} \end{bmatrix}\begin{bmatrix} {d_{1}(t)} \\ {d_{2}(t)} \end{bmatrix}} + {\begin{bmatrix} \theta_{5} & \theta_{6} \\ \theta_{7} & \theta_{8} \end{bmatrix}\begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}}};} & \left( {{Eq}.\mspace{11mu} U} \right) \end{matrix}$

or compactly as:

$\begin{matrix} {{\begin{bmatrix} {\overset{\cdot}{d}}_{1} \\ {\overset{\cdot}{d}}_{2} \end{bmatrix} = {{{A_{d}\begin{bmatrix} d_{1} \\ d_{2} \end{bmatrix}} + {B_{dd}u}} = {{\begin{bmatrix} \theta_{1} & \theta_{2} \\ \theta_{3} & \theta_{4} \end{bmatrix}\begin{bmatrix} d_{1} \\ d_{2} \end{bmatrix}} + {\begin{bmatrix} \theta_{5} & \theta_{6} \\ \theta_{7} & \theta_{8} \end{bmatrix}\begin{bmatrix} T_{sp} \\ T_{oa} \end{bmatrix}}}}};} & \left( {{Eq}.\mspace{11mu} U} \right) \end{matrix}$

where θ₁ through θ₈ are parameters that can be identified or set to a prospected value, and all other variables being the same as above. During step 1908, a Kalman gain for the augmented system is determined (or identified). The Kalman gain and the augmented system matrices can be used together to estimate historical heat disturbance values. In some embodiments, step 1908 is performed by controller 212.

Process 1900 is shown to include estimating historical heat disturbances using the disturbance model and the training data (step 1910), according to some embodiments. To estimate the historical heat disturbance, the disturbance model including the augmented system matrices as well as the Kalman gain can use the training data as input. In general, accuracy of the estimated historical heat disturbance hinges on accuracy of the system identification in step 1904 and parameters of the disturbance model identified/selected in step 1908. Accordingly, one result of process 1900 is a set of estimated historical heat disturbance values that includes a heat disturbance value for a plurality of time steps for which training data is available. In some embodiments, step 1910 is performed by controller 212.

Referring now to FIG. 20, a process 2000 for predicting a forecasted heat disturbance {circumflex over (Q)}_(other) _(forecast) and performing model predictive control based on the forecasted heat disturbance is shown, according to some embodiments. Process 2000 uses an autoregressive (AR) model to model a stochastic heat disturbance. In some embodiments, the AR model is used online (i.e., while the system is operating) to correct predictions of a deterministic heat disturbance by accounting for residuals (i.e., prediction errors), thereby predicting the stochastic heat disturbance of Q_(other) . In some embodiments, some and/or all steps of process 2000 are performed by controller 212.

Process 2000 is shown to include estimating historical heat disturbance (step 2002), according to some embodiments. In some embodiments, step 2002 is accomplished by performing process 1900 described with reference to FIG. 19. Process 2000 may require the system matrices, the disturbance model, the Kalman gain, and the estimated historic heat disturbance as determined in process 1900 to estimate {circumflex over (Q)}_(other) _(forecast) . In some embodiments, step 2002 is performed by controller 212.

Process 2000 is shown to include receiving an augmented system model, a disturbance model, and estimated historical heat disturbances (step 2004), according to some embodiments. In general, the disturbance model and the estimated historical heat disturbances are as determined in step 2002. In some embodiments, step 2004 is performed by controller 212.

Process 2000 is shown to include training a deterministic heat disturbance model for online predictions of the deterministic portion of heat disturbances (step 2005), according to some embodiments. In some embodiments, the deterministic heat disturbance model uses a linear regression and data fitting and/or pattern recognition to estimate the deterministic heat disturbance value similar to and/or the same as a deterministic load as described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, incorporated by reference herein in its entirety. The disturbance model can be trained using the training data of process 1900 and the estimated historical heat disturbance so that the disturbance model can properly estimate deterministic heat disturbance values based on environmental conditions, a time of day, a day of the week, etc. In some embodiments, step 2005 is performed by controller 212.

Process 2000 is shown to include training an autoregressive (AR) model to predict residuals of the disturbance model (step 2006), according to some embodiments. In some embodiments, the AR model is trained using the estimated historical heat disturbances gathered in step 2002. In some embodiments, the AR model is a first order model that captures the stochastic part of the heat disturbance Q_(other) . In some embodiments, the AR model is a higher order model. The AR model can be determined off-line and may not be part of a state space model used during a model predictive control process. In general, a first order AR model can have the following form:

ê(k+1)=a ₁ e(k)   (Eq.V);

where

e(k)=Q _(other) _(measured) (k)−{circumflex over (Q)} _(other) _(deterministic) (k)

where k is a time step, ê(k+1) is a residual for a next time step, a₁ is a constant, Q_(other) _(measured) (k) is measurement of the heat disturbance for time step k, and {circumflex over (Q)}_(other) _(deterministic) (k) is the deterministic heat disturbancefor time step k. As Q_(other) _(measured) (k) cannot be directly obtained, Q_(other) _(measured) (k)=d₁(k) as estimated in Eq. P and Eq. R. As a result, a residual can be determined for a current time step and estimated for future time steps by the AR model. In some embodiments, the AR model trained to predict residuals is similar to and/or the same as an autoregressive model to predict residuals as described in U.S. patent application Ser. No. 14/717,593, filed May 20, 2015, incorporated by reference herein in its entirety. In some embodiments, step 2006 is performed by controller 212.

Process 2000 is shown to include collecting current environmental data (step 2008), according to some embodiments. The current environmental data can include, for example, temperature measurements, humidity measurements, external weather conditions, air quality, etc. The current environmental data can be collected by various sensors in a building, a weather service, and/or any device/system capable of collecting and communicating environmental data to controller 212. For example, temperature sensor 214 and temperature sensor 216 may collect and communicate an indoor air temperature and an outdoor air temperature to controller 212 respectively. The current environmental data may also include information that can result in a heat disturbance such as, for example, occupancy in a building, current electrical consumption, time of day, etc. In some embodiments, step 2008 is performed by controller 212 and various devices/services capable of collecting and communicating environmental data (e.g., temperature sensor 214).

Process 2000 is shown to include performing an online state estimation using the system matrices and the current environmental data (step 2010), according to some embodiments. In step 2010, the five state model described in Eq. P and Eq. R can be converted to discrete time. Further, a state estimation can be conducted using the Kalman gain identified for the model. For each time step k, a state vector of the following form can be calculated:

$\quad\begin{bmatrix} {T_{ia}(k)} \\ {T_{m}(k)} \\ {I(k)} \\ {d_{1}(k)} \\ {d_{2}(k)} \end{bmatrix}$

where all variables are the same as described in Eq. P and Eq. R for each time step k. In some embodiments, step 2010 is performed by controller 212.

Process 2000 is shown to include estimating a deterministic heat disturbance value for a current time step using the disturbance model and the current environmental data (step 2012), according to some embodiments. Specifically, the current environmental data can be applied to the disturbance model to obtain the estimation of the deterministic heat disturbance value. In some embodiments, step 2012 is performed by controller 212.

Process 2000 is shown to include calculating a current residual based on the online state estimation and the estimated deterministic heat disturbance for the current time step (step 2014), according to some embodiments. In general, the current residual can be calculated by the following equation:

e(k)=d ₁(k)−{circumflex over (Q)} _(other) _(deterministic) (k)   (Eq. W)

where e(k) is the current residual, d₁(k) is a value of heat disturbance for the current time step determined based on the online state estimation performed in step 2010, and {circumflex over (Q)}_(other) _(deterministic) (k) is the deterministic heat disturbance estimated in step 2012. The current residual can illustrate the inaccuracy of {circumflex over (Q)}_(other) _(deterministic) (k) due to the stochastic heat disturbance. In general, a large value (negative or positive) of the current residual indicates {circumflex over (Q)}_(other) _(deterministic) (k) is inaccurate while values of the current residual close to 0 indicate {circumflex over (Q)}_(other) _(deterministic) (k) is more accurate. If the current residual is equal to 0, the stochastic heat disturbance may have no effect on heat disturbance. In some embodiments, step 2014 is performed by controller 212.

Process 2000 is shown to include predicting a stochastic heat disturbance for subsequent time steps using the autoregressive (AR) model and the current residual (step 2016), according to some embodiments. As the current residual can indicate a stochastic heat disturbance for a current time step, the current residual can be used as input to the AR model for predicting the stochastic heat disturbance for the subsequent time steps (i.e., the model trained at step 2006). Based on a magnitude of the stochastic heat disturbance, the AR model can determine an accuracy of the estimated deterministic heat disturbance for the current time step. As the magnitude of the stochastic heat disturbance grows, additional correction may be necessary by the AR model to predict an accurate stochastic heat disturbance for the subsequent time steps. In some embodiments, the stochastic heat disturbance for a next time step is described as {circumflex over (Q)}_(other) _(stochastic) (k+1) where k+1 is the next time step (i.e., the time step after time step k). Similarly, the stochastic heat disturbance for any subsequent time steps can be described as {circumflex over (Q)}_(other) _(stochastic) (k+n) for some n^(th) subsequent time step. In some embodiments, step 2016 is performed by controller 212.

Process 2000 is shown to include predicting a deterministic heat disturbance for the subsequent time steps using the disturbance model and the current environmental data (step 2018), according to some embodiments. The disturbance model can be configured to estimate future deterministic heat disturbances based on current data. As such, the current environmental data can be used as input to the disturbance model such that the disturbance model can output the predicted deterministic heat disturbance for the subsequent time steps. When predicting the deterministic heat disturbance for the subsequent time steps, the disturbance model may not use the stochastic heat disturbance for the subsequent time steps predicted in step 2016. As such, step 2018 may be similar to step 2012 for the subsequent time steps. In general, the deterministic heat disturbance for a next time step is described as {circumflex over (Q)}_(other) _(deterministic) (k+1) where k+1 is the next time step. Similarly, the deterministic heat disturbance for any subsequent time step can be described as {circumflex over (Q)}_(other) _(deterministic) (k+n) for some n^(th) subsequent time step. In some embodiments, step 2018 is performed by controller 212.

Process 2000 is shown to include predicting a forecasted heat disturbance for the subsequent time steps as a sum of the deterministic heat disturbance and the stochastic heat disturbance (step 2020), according to some embodiments. The forecasted heat disturbance {circumflex over (Q)}_(other) _(forecast) (k+1) is an estimated value of total heat disturbance for the next time step k+1. In general {circumflex over (Q)}_(other) _(forecast) (k+1) can be predicted by the following equation:

{circumflex over (Q)} _(other) _(forecast) (k+1)={circumflex over (Q)}_(other) _(deterministic) (k+1)+{circumflex over (Q)} _(other) _(stochastic) (k+1)   (Eq. X);

or as:

{circumflex over (Q)}_(other) _(forecast) (k+1)={circumflex over (Q)}_(other) _(deterministic) (k+1)+ê(k+1)   (Eq. Y);

where all variables are as defined above. If {circumflex over (Q)}_(other) _(forecast) (k+1) is a positive value, the predicted heat disturbance for the next time step may be adding additional heat into a space (e.g., building 10). If {circumflex over (Q)}_(other) _(forecast) (k+1) is a negative value, the predicted heat disturbance for the next time step may be taking heat out of the space. {circumflex over (Q)}_(other) _(forecast) can similarly be determined for other subsequent time steps based on {circumflex over (Q)}_(other) _(deterministic) and {circumflex over (Q)}_(other) _(stochastic) for each subsequent-time step. In some embodiments, step 2020 is performed by controller 212.

Process 2000 is shown to include performing a model predictive control process to control building equipment using the forecasted heat disturbance and the identified model (step 2022), according to some embodiments. By taking into account the forecasted heat disturbance, the model predictive control process can further optimize (e.g., reduce) costs related to operation of building equipment. For example, if the forecasted heat disturbance is positive and a building zone requires heating to maintain occupant comfort, the model predictive control process may determine that a heater is not required to be operated as the heat disturbance will increase a temperature of the building zone regardless. Without accounting for the forecasted heat disturbance, the model predictive control process may otherwise make control decisions that do not maintain occupant comfort and/or do not optimize costs. In some embodiments, step 2022 is performed by controller 212.

Referring now to FIG. 21, a process 2100 for determining a deterministic heat disturbance and performing a model predictive control process based on the deterministic heat disturbance is shown, according to some embodiments. Process 2100 utilizes multistep system identification to determine a Kalman gain and a stochastic disturbance model that is part of a state space system. In process 2100, the heat disturbance predictions from the deterministic heat disturbance model can be used as input for determining overall heat disturbance. Further, the Kalman gain and the stochastic disturbance model can be identified to account for inaccuracy in prediction of the deterministic heat disturbance due to inherent randomness, thereby allowing a stochastic heat disturbance to be calculated. In some embodiments, some and/or all steps of process 2000 are performed by controller 212.

Process 2100 is shown to include estimating historical heat disturbance (step 2102), according to some embodiments. In some embodiments, step 2102 is accomplished by performing process 1900 described with reference to FIG. 19. In some embodiments, step 2102 is performed by controller 212.

Process 2100 is shown to include training a deterministic heat disturbance model for online predictions of the deterministic portion of heat disturbances (step 2104), according to some embodiments. In some embodiments, the deterministic heat disturbance model uses a linear regression and data fitting and/or pattern recognition to estimate the deterministic heat disturbance value similar to and/or the same as a deterministic load as described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, incorporated by reference herein in its entirety. The disturbance model can be trained using the training data of process 1900 and the estimated historical heat disturbance so that the disturbance model can properly estimate deterministic heat disturbance values based on environmental conditions. In some embodiments, step 2104 is performed by controller 212.

Process 2100 is shown to include receiving a system model and augmenting the system model with a stochastic state space model and a Kalman gain (step 2106), according to some embodiments. Specifically, the stochastic state space model used to augment the system model can be a new disturbance model A_(d) _(e) that captures the stochastic piece of the heat disturbance. In general, the augmented system model can have the following form:

$\begin{matrix} {\begin{bmatrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ \overset{.}{I} \\ {\overset{.}{d}}_{e} \end{bmatrix} = {\quad{\begin{bmatrix} {\frac{1}{C_{ia}}\left( {{- K_{p}} - \frac{1}{R_{mi}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{mi}} & \frac{K_{I}}{C_{ia}} & \frac{1}{C_{ia}} \\ \frac{1}{C_{m}R_{mi}} & {- \frac{1}{C_{m}R_{mi}}} & 0 & 0 \\ {- 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & A_{d_{e}} \end{bmatrix}{\quad{{\begin{bmatrix} T_{ia} \\ T_{m} \\ I \\ d_{e} \end{bmatrix} + {\begin{bmatrix} \frac{K_{p}}{C_{ia}} & \frac{1}{C_{ia}R_{oi}} & \frac{1}{C_{ia}} \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} T_{sp} \\ T_{oa} \\ {\hat{Q}}_{{other}_{deterministic}} \end{bmatrix}}};}}}}} & \left( {{Eq}.\mspace{11mu} Z} \right) \\ {\begin{bmatrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{Q}}_{HVAC} \end{bmatrix} = {\begin{bmatrix} 1 & 0 & 0 & 0 \\ {- K_{p}} & 0 & K_{I} & 0 \end{bmatrix}{\quad{{\begin{bmatrix} T_{ia} \\ T_{m} \\ I \\ d_{e} \end{bmatrix} + {\begin{bmatrix} 0 & 0 & 0 \\ K_{p} & 0 & 0 \end{bmatrix}\begin{bmatrix} T_{sp} \\ T_{oa} \\ {\hat{Q}}_{{other}_{deterministic}} \end{bmatrix}}};}}}} & \left( {{Eq}.\mspace{11mu} {AA}} \right) \end{matrix}$

where A_(d) _(e) is the new disturbance model, d_(e) is a disturbance state that captures the stochastic heat disturbance {circumflex over (Q)}_(other) _(deterministic) is the deterministic heat disturbance, and all other variables are as defined above. For the augmented system model to be usable, values for the Kalman gain and A_(d) _(e) must be identified as described in greater detail below in step 2108. In some embodiments, the disturbance states at each time step d_(e)(k) only capture the stochastic heat disturbance as the deterministic heat disturbance is already accounted for. In some embodiments, the above four state model is used for both online estimations and online model predictive control (MPC) predictions. Further, the state space system above can be converted to a discrete time model to estimate heat disturbance at particular points in time over a time horizon. In some embodiments, step 2106 is performed by controller 212.

Process 2100 is shown to include determining an identified model by performing a system identification to determine values of the Kalman gain and stochastic disturbance model parameter(s) based on the historical heat disturbances and the training data (step 2108), according to some embodiments. Based on the historical heat disturbances, the Kalman gain can be determined such that the Kalman gain accounts for previous heat disturbances during previous time steps. By accounting for the historical heat disturbances, the Kalman gain (i.e., an adjustment for how inaccurate the deterministic heat disturbance is due to the stochastic heat disturbance) can be more precisely estimated to account for actual dynamics of the system. In step 2108, values of A_(d) _(e) (i.e., values of the stochastic disturbance model parameter(s)) can also be determined as part of the performed system identification. In some embodiments, multi-step ahead prediction is utilized in identification of A_(d) _(e) and the Kalman gain for improved estimation of the stochastic heat disturbance. In some embodiments, step 2108 is performed by controller 212.

Process 2100 is shown to include collecting an environmental condition forecast and/or a weather forecast (step 2110), according to some embodiments. The environmental condition forecast can include current and/or predicted environmental conditions for future times. The environmental condition forecast can include, for example, current/predicted temperature measurements, current/predicted humidity measurements, current/predicted external weather conditions, current/predicted air quality, etc. The environmental condition forecast can be collected by various sensors in a building, a weather service, and/or any device/system capable of collecting and communicating environmental data to controller 212. For example, temperature sensor 214 and temperature sensor 216 may collect and communicate an indoor air temperature and an outdoor air temperature to controller 212 respectively. The environmental condition forecast may also include information regarding factors that can result in a heat disturbance such as, for example, occupancy in a building, current electrical consumption, time of day, etc. The weather forecast collected in step 2110 can include predictions of external weather conditions at future times. The weather forecast can be obtained by, for example, requesting the weather forecast from an application programming interfaces (APIs) that provides weather forecasts to requesting services. The weather forecast can be utilized for estimating heat disturbances due to external conditions. In some embodiments, the environmental condition forecast and the weather forecast are part of a single forecast. In some embodiments, step 2110 is performed by controller 212 and various devices/services capable of collecting and communicating environmental data (e.g., temperature sensor 214).

Process 2100 is shown to include predicting deterministic heat disturbances using the disturbance model along with the environmental condition forecast and/or the weather forecast (step 2112), according to some embodiments. The deterministic heat disturbances may need to be estimated as the deterministic heat disturbance is used as input to the disturbance model for predicting the stochastic heat disturbances. In some embodiments, step 2112 is similar to and/or the same as step 2012 described with reference to FIG. 20. In general, the disturbance model can utilize the environmental condition forecast and/or the weather forecast as input and output a deterministic heat disturbance prediction. In some embodiments, the disturbance model utilizes the entirety of the environmental condition forecast and/or the weather forecast as input. In some embodiments, the disturbance model utilizes a portion of the environmental condition forecast and/or the weather forecast as input. In some embodiments, step 2112 is performed by controller 212.

Process 2100 is shown to include performing a model predictive control process to control building equipment using the deterministic heat disturbance predictions and the identified model (step 2114), according to some embodiments. Based on results of step 2112, the deterministic heat disturbance predictions can be used as input to the identified model to predict the stochastic heat disturbances. The deterministic heat disturbance predictions and the stochastic heat disturbance predictions can be used to determine a total heat disturbance to be considered during the model predictive control process. By taking into account the total heat disturbance, the model predictive control process can further optimize (e.g., reduce) costs related to operation of building equipment operable to heat or cool a building. For example, if the total heat disturbance is positive and a building zone requires heating to maintain occupant comfort, the model predictive control process may determine that a heater is not required to be operated as the heat disturbance will increase a temperature of the building zone regardless. Without accounting for the total heat disturbance, the model predictive control process may otherwise make control decisions that do not maintain occupant comfort and/or do not optimize costs. In some embodiments, step 2114 is performed by controller 212.

Heat Disturbance Estimation and Prediction Overview

Referring generally to FIGS. 22-30, methods for estimating a heat disturbance {dot over (Q)}_(other) for use in performing a control process (e.g., MPC) is shown, according to some embodiments. As described above, an inaccurate disturbance model, quantized sensor measurements, and/or high frequency input characteristics can result in inaccurate heat disturbance estimations if not accounted for. As such, FIGS. 22-30 illustrate how these issues can be addressed to ensure accurate heat disturbance estimations can be obtained for use in control processes such as MPC.

Processes for Estimating Heat Disturbances

Referring now to FIG. 22, a process 2200 to training a deterministic model of heat disturbance is shown, according to some embodiments. In some embodiments, some and/or all steps of process 2200 are performed by controller 212 as described with reference to FIG. 2.

Process 2200 is shown to include obtaining zone group data describing thermal dynamics of a zone group (step 2202). To accurately train a deterministic model of heat disturbance, a minimum amount of data can be required. For example, step 2202 may include gathering a minimum of two weeks' worth of training data. During data gathering, step 2202 may include ensuring that specific data is obtained for use in training the deterministic model. For example, step 2202 can include gathering data such as an outdoor relative humidity (e.g., as a percentage), a dry bulb weather (e.g., in degrees Kelvin), a temperature of a zone (e.g., in degrees Kelvin), measurements of {dot over (Q)}_(HVAC) (e.g., in kilowatts), a time stamp associated with each data point, and a sampling rate of the data (i.e., how often data is collected).

In step 2202, the sampling rate of data can be set to an appropriate value to ensure a sufficient amount of data is gathered over a data gathering period (also referred to herein as a learning period). For example, the sampling rate can be set to sample data every five minutes. The zone group data gathered over the learning period can be sampled at a constant rate (e.g., every five minutes) to ensure that the data does not include variations due to differences in the sampling rate. Further, to ensure data representative of system dynamics is gathered, the learning period may be set to start and end at a same time on different days (e.g., from midnight to midnight). In some embodiments, step 2202 is performed by controller 212.

Process 2200 is shown to include performing a linear regression to estimate a thermal resistance between indoor air and outdoor air (step 2204). The thermal resistance between the indoor air and the outdoor air can be defined as

$\frac{1}{R_{oi}}.$

The linear regression performed in step 2204 can help in estimating heat transfer effects from outside air temperature. In performing the linear regression, {dot over (Q)}_(other) can be modeled as a constant offset. Further, the following zone temperature steady state equation can be utilized in performing the linear regression:

$0 = {{\frac{1}{R_{oi}}\left( {T_{oa} - T_{ia}} \right)} + {\overset{.}{Q}}_{HVAC} + {\overset{.}{Q}}_{other}}$

In this example of the zone temperature steady state equation, effects of building mass dynamics are not included.

To perform the linear regression, values of {dot over (Q)}_(HVAC) can be extracted from the zone group data to create a first column vector containing the values. The first column vector can have the following form:

$\quad\begin{bmatrix} {\overset{.}{Q}}_{HVAC_{1}} \\ {\overset{.}{Q}}_{HVAC_{2}} \\ \vdots \\ {\overset{.}{Q}}_{HVAC_{n}} \end{bmatrix}$

where {dot over (Q)}_(HVAC) _(x) indicates a value of {dot over (Q)}_(HVAC) at a time x and where n indicates a last recorded value of {dot over (Q)}_(HVAC). Likewise, a second column vector containing values of (T_(oa)−T_(ia)) of the zone temperature steady state equation can be generated to have the following form:

$\quad\begin{bmatrix} {T_{oa_{1}} - T_{ia_{1}}} \\ {T_{oa_{2}} - T_{ia_{2}}} \\ \vdots \\ {T_{oa_{n}} - T_{ia_{n}}} \end{bmatrix}$

where T_(oa) _(x) −T_(ia) _(x) indicates a value of T_(oa)−T_(ia) at time x.

To ensure accurate relationships between operation of HVAC equipment and the heat disturbance are maintained, any data points in the first column vector where {dot over (Q)}_(HVAC) equals zero can be removed as well as any corresponding data points from the second column vector (i.e., data points with a same timestamp as those removed from the first column vector). Removal of said data points can result in a modified first column vector of the following form:

$\quad\begin{bmatrix} {\overset{.}{Q}}_{HVAC_{1}} \\ {\overset{.}{Q}}_{HVAC_{2}} \\ \vdots \\ {\overset{.}{Q}}_{HVAC_{m}} \end{bmatrix}$

where m indicates a new last value of {dot over (Q)}_(HVAC) such that m≤n. It should be understood that {dot over (Q)}_(HVAC) ₁ and/or {dot over (Q)}_(HVAC) ₂ of the first column vector may or may not be the same as {dot over (Q)}_(HVAC) ₁ and/or {dot over (Q)}_(HVAC) ₂ of the modified first column vector depending on whether {dot over (Q)}_(HVAC) ₁ and/or {dot over (Q)}_(HVAC) ₂ of the first column vector are equal to zero. Likewise, a modified second column vector can be determined to have the form:

$\quad\begin{bmatrix} {T_{oa_{1}} - T_{ia_{1}}} \\ {T_{oa_{2}} - T_{ia_{2}}} \\ \vdots \\ {T_{oa_{m}} - T_{ia_{m}}} \end{bmatrix}$

where values of T_(oa) _(x) −T_(ia) _(x) from the second column vector are removed if {dot over (Q)}_(H{dot over (Q)}HVAC) _(x) =0 for some time x.

Based on the modified first column vector without data points where {dot over (Q)}_(HVAC) equals zero, a third column vector can be generated. The third column vector can be generated such that the third column vector is of the same length as the modified first column vector and includes only the value 1. In other terms, the third column vector can be represented as:

$\quad\begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}$

where a length of the third column vector is the same as the length of the modified first column vector corresponding to the modified first column vector.

Further, a two column matrix A can be generated where A is a combination of the modified second column vector and the third column vector. In particular, a first column of A can include values of (T_(oa)−T_(ia)) excluding those removed that have the same timestamp as data points where {dot over (Q)}_(HVAC) equals zero. Likewise, a second column of A can be of the same length as the third column vector and include all 1's. As such, A can have the following form:

$A = \begin{bmatrix} {T_{oa_{1}} - T_{ia_{1}}} & 1_{1} \\ {T_{oa_{2}} - T_{ia_{2}}} & 1_{2} \\ \vdots & \vdots \\ {T_{oa_{m}} - T_{ia_{m}}} & 1_{m} \end{bmatrix}$

Further, in performing the linear regression, a vector b can be generated such that b=−{dot over (Q)}_(HVAC). As such, b can have the following form:

$b = \begin{bmatrix} {- {\overset{.}{Q}}_{HVAC_{1}}} \\ {- {\overset{.}{Q}}_{HVAC_{2}}} \\ \vdots \\ {- {\overset{.}{Q}}_{HVAC_{m}}} \end{bmatrix}$

Based on A and b, an equation Ax=b can be solved by using a pseudoinverse computed as the following:

x=(A ^(T) A)⁻¹ A ^(T) b

where the T superscript indicates a transposed matrix. In this way, a first element of the x vector can define

$\frac{1}{R_{oi}}.$

In some embodiments, step 2204 is performed by controller 212.

Process 2200 is shown to include computing an initial profile of heat disturbance base on the thermal resistance between the indoor air and the outdoor air (step 2206). As {dot over (Q)}_(other) is modeled as a constant offset in the linear regression performed in step 2204,

$\frac{1}{R_{oi}}$

may be an approximation. Based on the approximate value of

$\frac{1}{R_{oi}},$

the initial protile of {dot over (Q)}_(other) (referred to as {dot over (Q)}_(other) _(Base)) can be calculated using the zone temperature steady state equations not including the building mass temperature effects. In general, {dot over (Q)}_(other) _(Base) can be calculated by the following equation:

${\overset{.}{Q}}_{other_{Base}} = {{{- \frac{1}{R_{oi}}}\left( {T_{oa} - T_{ia}} \right)} - {\overset{.}{Q}}_{HVAC}}$

In some embodiments, step 2206 is performed by controller 212.

Process 2200 is shown to include filtering the initial profile of heat disturbance (step 2208). If generated {dot over (Q)}_(other) _(Base) may ,include various noise and spikes that can lead to inaccuracies of predictions and/or other determinations made based on {dot over (Q)}_(other) _(Ba) se As such, to ensure {dot over (Q)}_(other) _(Base) includes accurate estimations of a magnitude of {dot over (Q)}_(other) for each day {dot over (Q)}_(other) _(Base) can be filtered to reduce an amount of noise and spikes. To filter {dot over (Q)}_(other) _(Base) , {dot over (Q)}_(other) _(Base) can be passed through an anti-spike filter to remove spikes in {dot over (Q)}_(other) estimations. Examples of anti-spike filters can include median filters, clamp filters, etc. {dot over (Q)}_(other) _(Base) can also be passed through a smoothing filter to reduce an impact of noise on estimations of {dot over (Q)}_(other) Examples of smoothing filters can include an exponentially-weighted moving average filter, a Savitzky-Golay filter, etc. In some embodiments, {dot over (Q)}_(other) _(Base) is not passed through any filters if a determination is made that {dot over (Q)}_(other) _(Base) does not include excessive noise and/or spikes. In some embodiments, {dot over (Q)}_(other) _(Base) is only passed through one of the anti-spike filter or the smoothing filter. In some embodiments, {dot over (Q)}_(other) _(Base) is passed through additional filters and/or is passed through the anti-spike filter and/or the smoothing filter multiple times. In other words, step 2208 can include passing {dot over (Q)}_(other) _(Base) through any necessary filters to get {dot over (Q)}_(other) _(Base) to include good (i.e., accurate) estimates of {dot over (Q)}_(other) . In some embodiments, step 2208 is performed by controller 212.

Process 2200 is shown to include training a deterministic model of heat disturbance based on the filtered initial profile of heat disturbance (step 2210). In some embodiments, {dot over (Q)}_(other) _(Base) is used to train an approximate heat disturbance profile {dot over (Q)}_(other) _(Approx) . If {dot over (Q)}_(other) _(Base) is used to train {dot over (Q)}_(other) _(Approx) , {dot over (Q)}_(other) _(Approx) can be used to train the deterministic model. Generation of {dot over (Q)}_(other) _(Approx) is described in greater detail below with reference to FIG. 23. As such, step 2210 may include performing some and/or all of a process 2300 described with reference to FIG. 23.

Alternatively, the deterministic model can be trained directly based on filtered {dot over (Q)}_(other) _(Base) . If {dot over (Q)}_(other) _(Base) is used to directly train the deterministic model, processing efficiency of performing process 2200 may increase as fewer approximations and other calculations related to the heat disturbance may be needed as compared to generating {dot over (Q)}_(other) _(Approx) in order to train the deterministic model. However, {dot over (Q)}_(other) _(Approx) may provide a more accurate deterministic model, thereby resulting in more accurate heat disturbance predictions and better control decisions that more precisely operate building equipment to account for the heat disturbance. As such, step 2210 may include determining whether to train the deterministic model directly based on {dot over (Q)}_(other) _(Base) or if to first generate {dot over (Q)}_(other) _(Approx) and use {dot over (Q)}_(other) _(Approx) totrain the deterministic model. In some embodiments, step 2210 is performed by controller 212.

Referring now to FIG. 23, process 2300 for training a deterministic model of heat disturbance based on an initial profile of the heat disturbance is shown, according to some embodiments. In some embodiments, process 2300 is performed based on a determination in step 2210 of process 2200 as described with reference to FIG. 22 that the deterministic model is to be trained based on an approximate profile of {dot over (Q)}_(other), referred to as {dot over (Q)}_(other) _(Approx) . In some embodiments, some and/or all steps of process 2300 are performed by controller 212.

Process 2300 is shown to include obtaining an initial profile of heat disturbance (step 2302). As described above with reference to FIG. 22, the initial profile of heat disturbance can be referred to {dot over (Q)}_(other) _(Base) . In some embodiments, step 2302 is completed by performing some and/or all of process 2200 as described with reference to FIG. 22. In some embodiments, step 2302 is performed by controller 212.

Process 2300 is shown to include fitting each day's worth of the initial profile to a predefined function form to obtain an approximate profile of the heat disturbance (step 2304). {dot over (Q)}_(other) _(Base) can include accurate or relatively accurate estimates of a magnitude of {dot over (Q)}_(other) in each day. However, {dot over (Q)}_(other) _(Base) may not include a correct shape of {dot over (Q)}_(other) if a high frequency {dot over (Q)}_(HVAC) input signal is present. The shape of {dot over (Q)}_(other) can be defined by how estimates of {dot over (Q)}_(other) determined based on {dot over (Q)}_(other) _(Base) change over each day. The high frequency {dot over (Q)}_(HVAC) input signal can result in difficulty in distinguishing high frequency noise from an actual signal, thereby leading to a loss of signal contents even if {dot over (Q)}_(other) _(Base) is filtered. Advantageously, effects of the high frequency {dot over (Q)}_(HVAC) input signal can be mitigated by fitting each day's worth of {dot over (Q)}_(other) _(Base) data to a predefined function form. The total of each fitted profile can be used to define {dot over (Q)}_(other) _(Approx) .

The function forms used to fit each day's worth of {dot over (Q)}_(other) _(Base) data can be chosen based on an expected heat load profile of an associated building (e.g., building 10). Different types of buildings may have varying expected heat load profiles based on a purpose of a building. For example, a residential building may have a different expected heat load profile than a commercial building as times when occupants are present may be different, usage of electronics may differ, etc. To obtain {dot over (Q)}_(other) _(Approx) , various predefined function forms can be used. In particular, the function forms can include, for example, a Gaussian function, a sine function, and a user defined function, all described in greater detail below. In some embodiments, a different function form is used rather than the Gaussian function, the sine function, or the user defined function to obtain {dot over (Q)}_(other) _(Approx) . The Gaussian function, the sine function, and the user defined function are provided purely for sake of example.

The Gaussian function can be defined by a target daily fitting function shown as:

${f(x)} = {{a \cdot e^{- \frac{{({x - b})}^{2}}{2\; c^{2}}}} + d}$

where a is an amplitude factor where a∈[−∞, ∞], b is a mean or horizontal shift where b∈[0, samples in one day], c is a standard deviation where c∈[−∞, ∞], d is a vertical shift where d∈[−∞, ∞], and x is a sample index defined as a consecutive integer array starting from one to a total number of samples per day. In some embodiments, c can change shape to be wide or narrow as needed.

For each day, a separate f (x) fit can be found by determining optimal values of a, b, c, and d that minimize a two-norm error between f(x) and {dot over (Q)}_(other) _(Base) for a particular day. In general, a two-normal measure can be defined as the square root of the squares of each data point. For example, the two-norm of {dot over (Q)}_(other) _(Base) can be shown as:

${Twonorm}{\left( {\overset{.}{Q}}_{o{ther}_{Base}} \right) = \sqrt{{\overset{.}{Q}}_{{other}_{Base_{1}}}^{2} + {\overset{.}{Q}}_{o{ther}_{Base_{2}}}^{2} + \ldots + {\overset{.}{Q}}_{o{ther}_{Base_{n}}}^{2}}}$

where

${\overset{.}{Q}}_{{other}_{Base_{x}}}$

is an estimation of {dot over (Q)}_(other) based on {dot over (Q)}_(other) _(Base) for a time step x and n is a total number of time steps. Based on each f(x), a total of all f(x) can be determined to make up {dot over (Q)}_(other) _(Approx) . Possible shapes of the Gaussian function are described in greater detail below with reference to FIGS. 25A-C.

As for the sine function, the target daily fitting function can be shown as:

f(x)=a ·sin(b·(x−c))+d

where a is an amplitude factor where a∈[−∞, ∞], b is a period where b∈

$\left\lbrack {\frac{0.25\pi}{{samples}\mspace{14mu} {in}\mspace{14mu} {one}\mspace{14mu} {day}},\frac{2\pi}{{samples}\mspace{14mu} {in}\mspace{14mu} {one}\mspace{14mu} {day}}} \right\rbrack,$

c is a phase shift where c∈[0, samples in one clay], d is a vertical shift where d∈[−∞, ∞], and x is a sample index defined as a consecutive integer array starting from one to a total number of samples per day.

Similar to the Gaussian function, a separate f (x) can be found for each day by determining optimal values of a, b, c, and d that minimize the two-norm error between f(x) and {dot over (Q)}_(other) _(Base) for a particular day. Based on each f(x), a total of all f(x) can be determined to make up {dot over (Q)}_(other) _(Approx) .

As for the user defined function, a user can set their own function to generate {dot over (Q)}_(other) _(Approx) . An example of a target daily fitting function can be determined by generating a normalized {dot over (Q)}_(other) profile that has an expected shape with a peak around noon. An example of the normalized{dot over (Q)}_(other) profile is shown below in graph 2600 as described with reference to FIG. 26. Based on (filtered) {dot over (Q)}_(other) _(Base) profile obtained in step 2302, the normalized {dot over (Q)}_(other) profile can be set such that a peak of the normalized {dot over (Q)}_(other) profile can be set equal to a maximum value of {dot over (Q)}_(other) _(Base) for a particular day. In other words, the maximum value of {dot over (Q)}_(other) _(Base) can be multiplied by the normalized{dot over (Q)}_(other) profile for each day. Said multiplication can result in a scaled profile (i.e., {dot over (Q)}_(other) _(Approx) ) that has a unique scale for each day.

As mentioned above, the Gaussian function, the sine function, and the user defined function are given purely for sake of example. The present disclosure contemplates using various other functions to generate {dot over (Q)}_(other) _(Approx) . In some embodiments, step 2304 is performed by controller 212.

Process 2300 is shown to include training a deterministic model of the heat disturbance based on the approximate profile, a relative humidity, and an outdoor air temperature (step 2306). The relative humidity and the outdoor air temperature can be obtain through various sensors, services, etc. For example, the relative humidity and/or the outdoor air temperature can be obtained by outdoor air temperature sensor 216. As another example, the relative humidity and/or the outdoor air temperature can be obtained from a weather service that provides data regarding various environmental conditions. The relative humidity and the outdoor air temperature can be particularly helpful in training the deterministic model as the relative humidity and the outdoor air temperature can potentially impact the heat disturbance affecting a building. In some embodiments, other various environmental conditions (e.g., cloud cover) are used to train the deterministic model. In some embodiments, the deterministic heat disturbance model is obtain by performing a linear regression and data fitting process as described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, incorporated by reference herein in its entirety. In some embodiments, step 2306 is performed by controller 212.

Referring now to FIG. 24, a process 2400 for identifying a state space system for use in a model predictive control (MPC) process is shown, according to some embodiments. Process 2400 can be useful for obtaining accurate predictions of heat disturbance based on both a deterministic model and a stochastic model. Based on said predictions, a control process such as MPC can determine how to operate building equipment (e.g., HVAC equipment 210) over the duration of an optimization period in order to optimize (e.g., reduce) costs and/or maintain occupant comfort. By accounting for the heat disturbance, control decisions related to operation of the building equipment can result in desired changes to temperature and/or other environmental conditions in a building (e.g., building 10). In some embodiments, some and/or all steps of process 2400 are performed by controller 212.

Process 2400 is shown to include obtaining a deterministic model of heat disturbance (step 2402). In some embodiments, step 2402 includes performing process 2300 described with reference to FIG. 23 and/or process 2400 described with reference to FIG. 24. As such, the deterministic model obtained in step 2402 may be trained based on either {dot over (Q)}_(other) _(Base) or {dot over (Q)}_(other) _(Approx) . In some embodiments, step 2402 is performed by controller 212.

Process 2400 is shown to include predicting deterministic portion profiles of the heat disturbance based on the deterministic model (step 2404). The deterministic portion profiles can define predictions of the deterministic portion of {dot over (Q)}_(other) over a time period. As the deterministic model is trained to represent how {dot over (Q)}_(other) changes over time in a building based on varying conditions (e.g., relative humidity, outdoor air temperature, etc.), conditions can be supplied as input to the model to generate the deterministic portion profiles. In some embodiments, step 2404 is performed by controller 212.

Process 2400 is shown to include performing a system identification process to identify a state space system of a building thermal model, a Kalman gain, a stochastic heat disturbance model, and a scaling parameter of the heat disturbance deterministic portion profiles (step 2406). The building thermal model can describe various thermal dynamics of a building (e.g., building 10). In some embodiments, the building thermal model identified in the system identification has the following form:

${A(\theta)} = {{\begin{bmatrix} {- \left( {\theta_{1} + \theta_{2}} \right)} & \theta_{2} \\ \theta_{3} & {- \theta_{3}} \end{bmatrix}{B(\theta)}} = \begin{bmatrix} \theta_{4} & \theta_{1} & {\theta_{5}\theta_{4}} \\ 0 & 0 & 0 \end{bmatrix}}$ C(θ) = [1  0], D(θ) = [0  0]

where

${\theta_{1} = \frac{1}{C_{ia}R_{oi}}},{\theta_{2 =}\frac{1}{C_{ia}R_{mi}}},{\theta_{3} = \frac{1}{C_{m}R_{mi}}},{\theta_{4} = \frac{1}{C_{ia}}},$

and θ₅ is the scaling parameter of the heat disturbance deterministic portion profiles also referred to as {circumflex over (Q)}_(other) _(deterministic) Scale (i.e., θ₅={circumflex over (Q)}_(other) _(deterministic) Scale). As the deterministic model can be obtained assuming steady state operation, predictions based on the deterministic model may reflect the shape of {dot over (Q)}_(other), but not a scale of {dot over (Q)}_(other). As such, the scaling parameter can be identified such that predictions based on the deterministic model are closer to a true scale (i.e., true values) of {dot over (Q)}_(other). An example of the scaling parameter based on an experiment is shown in greater detail below with reference to FIG. 29. In some embodiments, the system identification is performed using a simulation prediction error method described in greater detail above with reference to FIG. 4 (i.e., to identify values of the parameters θ₁, θ₂, θ₃, θ₄, θ₅). In some embodiments, step 2406 is performed by controller 212.

Process 2400 is shown to include augmenting the state space system with a stochastic model of the heat disturbance and a Kalman gain identified by multi-step system identification (step 2408). In some embodiments, the stochastic model and/or the Kalman gain are identified in the system identification process performed in step 2406. In some embodiments, the stochastic model and/or the Kalman gain are identified in a separate multi-step system identification process performed in step 2408. The augmented state space model can be reflected by the following:

$\begin{bmatrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ {\overset{.}{d}}_{e} \end{bmatrix} = {\begin{bmatrix} {\frac{1}{C_{ia}}\left( {{- \frac{1}{R_{m\; i}}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{m\; i}} & \frac{1}{C_{ia}} \\ \frac{1}{C_{m}R_{m\; i}} & {- \frac{1}{C_{m}R_{m\; i}}} & 0 \\ 0 & 0 & A_{de} \end{bmatrix}{\quad{{\begin{bmatrix} T_{ia} \\ T_{m} \\ d_{e} \end{bmatrix} + {{\begin{bmatrix} \frac{1}{C_{ia}} & \frac{1}{C_{ia}R_{oi}} & {\theta_{5}\frac{1}{C_{ia}}} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} {\overset{.}{Q}}_{HVAC} \\ T_{oa} \\ {\hat{Q}}_{{Other}_{deterministic}} \end{bmatrix}}\mspace{20mu}\left\lbrack T_{ia} \right\rbrack}} = {{\left\lbrack {1\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack \begin{bmatrix} T_{ia} \\ T_{m} \\ d_{e} \end{bmatrix}} + {\left\lbrack {0\mspace{14mu} 0\mspace{14mu} 0} \right\rbrack \begin{bmatrix} {\overset{.}{Q}}_{HVAC} \\ T_{oa} \\ {\hat{Q}}_{{Other}_{deterministic}} \end{bmatrix}}}}}}$

where A_(d) _(e) is the stochastic model and all other variables are described above. If an auto-regressive (AR) model if chosen for the stochastic model, a full parameterized model for the system can be defined as:

${{A(\theta)} = \begin{bmatrix} {- \left( {\theta_{1} + \theta_{2}} \right)} & \theta_{2} & \theta_{4} \\ \theta_{3} & {- \theta_{3}} & 0 \\ 0 & 0 & {- \theta_{6}} \end{bmatrix}},{{B(\theta)} = \begin{bmatrix} \theta_{4} & \theta_{1} & {\theta_{5}\theta_{4}} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}}$ ${{C(\theta)} = \left\lbrack {1\ 0\ 0} \right\rbrack},,{{D(\theta)} = \left\lbrack {0\ 0\ 0} \right\rbrack},{{K(\sigma)} = \begin{bmatrix} \sigma_{1,1} \\ \sigma_{1,2} \\ \sigma_{1,3} \end{bmatrix}}$

where θ₆ is an AR coefficient and σ_(1,1), θ_(1,2), and θ_(1,3) are Kalman gain parameters. Values for θ₆ and σ_(1,1), σ_(1,2), and σ_(1,3) can be determined at step 2408. In some embodiments, step 2406 and step 2408 are part of a single step of process 2400. If step 2406 and step 2408 are part of a single step, the system identifications performed in step 2406 and step 2408 may be combined. In some embodiments, step 2408 is performed by controller 212.

Process 2400 is shown to include performing a model predictive control (MPC) process based on the deterministic model, the stochastic model, and the building thermal model to determine an optimal control strategy (step 2410). Effectively, by applying the state space model identified above, the heat disturbance can be determined. Based on the heat disturbance, MPC can determine how building equipment should be operated to optimize (e.g., reduce) costs and/or maintain occupant comfort. For example, if a space in a building is currently too hot for occupants and a heat disturbance due to solar radiation is present, MPC may determine that additional operation of building equipment is required to cool the space and offset effects of the heat disturbance. In general, by utilizing each model, MPC can generate modified control decisions that obtain desired adjustments to temperature of a space given heat disturbances affecting the space. The building equipment can be thereby controlled in accordance with an output of processes 2200, process 2300, and/or process 2400. In some embodiments, step 2410 is performed by controller 212.

Example Target Daily Fitting Functions

Referring generally to FIGS. 25A-26, example fitting functions that can be utilized by controller 212 to generate an approximate heat disturbance profile {dot over (Q)}_(other) _(Approx) are shown, according to some embodiments. In some embodiments, some and/or all of the functions of FIGS. 25A-26 are utilized in step 2304 of process 2300 as described with reference to FIG. 23.

Referring now to FIG. 25A, a graph 2500 of an estimated heat disturbance profile of a large zone group of a commercial building is shown, according to some embodiments. Graph 2500 is shown to include a series 2502. Series 2502 can illustrate how values a normalized {dot over (Q)}_(other) profile are expected to change over time for the large zone group of the commercial building. Series 2502 is shown to reach a maximum at approximately 12 hours (i.e., noon) of a day. As graph 2500 is relative to the commercial building, the heat disturbance can be expected to be large towards the middle of the day when most employees are expected to be working, solar radiation is near a maximum, etc. For example, since employees are typically expected to be at work near the middle of the day, a heat disturbance due to body heat and electronic equipment used by the occupants (e.g., computers, phones, etc.) may expected to be highest around noon.

Referring now to FIG. 25B, a graph 2504 of an estimated heat disturbance profile of a small zone group of a commercial building is shown, according to some embodiments. In some embodiments, graph 2504 is similar to graph 2500 as described with reference to FIG. 25A. Graph 2504 is shown to include a series 2506 illustrating values of a normalized {dot over (Q)}_(other) profile over a day for the small zone group. A maximum value of series 2506 is shown to be lower than a maximum value of series 2502 of graph 2500. Similarly, values of series 2506 are shown to change at a more gradual pace as compared to values of series 2502. As graph 2504 illustrates the small zone group, fewer effects of heat disturbance may impact the small zone group as compared to those impacting the large zone group illustrated by graph 2500. Similarly, as series 2506 illustrates normalized values, the heat disturbance may change more gradually as a maximum and a minimum heat disturbance are closer than as compared to the large zone group of graph 2500.

Referring now to FIG. 25C, a graph 2508 of an estimated heat disturbance profile of a small zone group of a residential building is shown, according to some embodiments. In some embodiments, graph 2508 is similar to graph 2500 and/or graph 2504 of FIGS. 25A and 25B respectively. Graph 2508 is shown to include a series 2510. Series 2510 can illustrate values of a normalized {dot over (Q)}_(other) profile over a day for the small zone group of the residential building. Unlike series 2502 or 2506, series 2510 is shown to have a minimum around noon rather than a maximum. As series 2510 represents a residential building rather than a commercial building, the expected profile of heat disturbance may decrease towards the middle of the day when occupants are expected to be gone (e.g., at work). Further, as occupants may expected to be away, the expected heat disturbance due to operation of electronic equipment may decrease as well. Also, as series 2510 applies to a residential building, expected variations in the heat disturbance are shown to be less than both series 2502 and 2506 due to, for example, fewer people occupying a residential building as compared to a commercial building.

Referring now to FIG. 26, a graph 2600 of a user defined target daily fitting function used for fitting an initial profile of heat disturbance is shown, according to some embodiments. Graph 2600 is shown to include a series 2602. Similar to series 2502, series 2506, and series 2510 of FIGS. 25A-C, series 2602 can be used to generate an approximate heat disturbance profile {dot over (Q)}_(other) _(Approx) . In particular, for each day's worth of data, the maximum value of {dot over (Q)}_(other) _(Base) can be multiplied by series 2602 to generate {dot over (Q)}_(other) _(Approx) . As series 2602 can be determined by a user, series 2602 may include various adjustments that reflect a building managed by the occupant. For example, as shown by series 2602, series 2602 has a maximum around 12 hours (i.e., noon), but also increases temporarily around 14 hours (i.e., 2:00 p.m.). The temporary increase around 14 hours may be defined by the user if, for example, the user expects a meeting in a zone defined by series 2602 to occur around 2:00 p.m. daily. In this way, {dot over (Q)}_(other) _(Approx) can be obtained such that {dot over (Q)}_(other) _(Approx) accounts for an additional heat disturbance around 2:00 p.m. due to the meeting. Of course, series 2602 is shown purely for sake of example. A user can define the normalized {dot over (Q)}_(other) profile to follow any various shape.

Experimental Results of Applying Heat Disturbance Estimations and Prediction to MPC

To illustrate the advantages of the processes described above, the following experimental results are included and shown in FIGS. 27-30. Particularly, the experimental results shown below are based on a two week data collection period.

Referring now to FIG. 27, a graph 2700 of a regression performed on data representing a correspondence between heat transfer from HVAC equipment and a temperature difference is shown, according to an example experiment. Graph 2700 is shown to include a regression line 2702. Regression line 2702 can be calculated by performing a linear regression based on various data points indicating a correspondence between a measurements of {dot over (Q)}_(HVAC) in kilowatts and a difference between a zone temperature and an outside air temperature in degrees Celsius represented by the various circles in graph 2700. In the example experiment illustrated in FIG. 27, a slope of regression line 2702 is equal to 0.824, thereby indicating

$\frac{1}{R_{oi}} = {{0.8}2{4.}}$

Based on

$\frac{1}{R_{oi}},$

{dot over (Q)}_(other) _(Base) can be calculated by the following:

${\overset{.}{Q}}_{o{ther}_{Base}} = {{{- \frac{1}{R_{oi}}}\left( {T_{oa} - T_{ia}} \right)} - {\overset{.}{Q}}_{HVAC}}$

{dot over (Q)}_(other) _(Base) can be passed through an anti-spike filter. In this example experiment, {dot over (Q)}_(other) _(Base) can be particularly passed through a median filter with a horizon looking forward and backward two samples as given by the following equation:

y(t)=median(u(t+2)+u(t+1)+u(t)+u(t−1)+u(t−2))

{dot over (Q)}_(other) _(Base) can be also passed through a smoothing filter. In the example experiment, {dot over (Q)}_(other) _(Base) can be particularly passed through a Savitzky-Golay filter with a first order polynomial and a horizon looking forward and backward 3 samples. In this way, a {dot over (Q)}_(other) _(Base) profile without excessive noise and/or spikes can be obtained.

Referring now to FIG. 28, a graph 2800 of an approximate heat disturbance profile over time is shown, according to an example experiment. Graph 2800 is shown to include a series 2802 illustrating values of {dot over (Q)}_(other) _(Approx) over time. {dot over (Q)}_(other) _(Approx) can be calculated based on the {dot over (Q)}_(other) _(Base) profile determined above with reference to FIG. 27 by taking a maximum value of the filtered {dot over (Q)}_(other) _(Base) for each day and scaling a normalized {dot over (Q)}_(other) profile to have peaks of the normalized {dot over (Q)}_(other) profile equal to the maximum value for each day. Series 2802 illustrates two weeks of {dot over (Q)}_(other) _(Approx) profile that can be used in conjunction with relative humidity and outside air temperature measurements to obtain a deterministic model of heat disturbance using linear regression and data fitting as described above with reference to FIG. 22.

Referring now to FIG. 29, a graph 2900 of deterministic heat disturbance predictions scaled by a scaling parameter is shown, according to some embodiments. Graph 2900 is shown to include a series 2902 illustrating deterministic predictions of the heat disturbance over a four day period determined based on the deterministic model described above with reference to FIG. 28. However, as {dot over (Q)}_(other) _(Base) may be obtained using steady state assumptions, series 2902 may include a correct shape of heat disturbances over time, but may not be scaled correctly. As such, graph 2900 is also shown to include a series 2904 and a series 2906. Series 2904 illustrates a situation in which series 2902 is scaled up to increase values of the deterministic predictions indicated by series 2902. Alternatively, series 2906 illustrates a situation in which series 2902 is scaled down to reduce values of the deterministic predictions.

To determine a proper scaling factor by which to scale series 2902, a system identification process can be performed to identify the scaling factor and a building thermal model. In the context of the example experiment illustrated by FIGS. 27-30, a scaling factor of 0.637 can be calculated to be an optimal scaling value by which to scale values of series 2902. As such, series 2906 can illustrate the scaled values of series 2902. It should be noted that series 2902-2904 may not be to scale.

Referring now to FIG. 30, a graph 3000, a graph 3002, a graph 3004, and a graph 3006 illustrating results of performing MPC to determine an optimal heating profile that maximizes cost savings while maintaining temperatures comfortable to occupants are shown, according to some embodiments. Based on the building thermal model and scaling parameter as described above with reference to FIG. 29, the building thermal model can be augmented with a stochastic disturbance model and a Kalman gain which can be identified through a multi-step system identification. In regards to the example experiment described in FIGS. 27-30, the multi-step horizon can be set to 72 steps (i.e., 6 hours for a sampling rate of 5 minutes). In this case, the stochastic disturbance model can be calculated to be A_(d) _(e) =−θ₆=−0.974.

By using the deterministic, stochastic, and building thermal models, MPC can be performed to generate control decisions that control building equipment in accordance with the estimated heat disturbance. In MPC, the control decisions can be generated to optimize (e.g., reduce) costs and/or maintain occupant comfort. In the context of the example experiment described in FIGS. 27-30, a comfortable range can be set such that a temperature of a space (e.g., a zone of a building) is to stay between 79° F. and 84° F.

Graph 3000 is shown to include a series 3008 illustrating outside air temperature over time. Tracking the outside air temperature may be crucial for accurately estimating the heat load disturbance. As the outside air temperature increases, a larger heat disturbance due to the outside air and/or solar irradiance may affect the space, thereby impacting how building equipment should be operated.

Graph 3002 is shown to include a series 3010, a series 3012, and a series 3014. Series 3010 illustrates a predicted building mass temperature of a building associated with the experiment. Series 3012 illustrates a predicted zone temperature for a zone of the building over time. Series 3014 illustrates a temperature setpoint generated based on MPC decisions.

Graph 3004 is shown to include a series 3016 illustrating an electricity rate profile over time. Series 3016 is shown to include two time periods at which a cost of electricity (as shown in dollars per kilowatt) is higher. The two time periods can illustrate peak consumption times at which an electricity supplier charges more for electricity. In order to generate cost-effective decisions, MPC may determine values of decision variables that reduce an amount of electricity consumed during the peak consumption times. For example, during the first peak consumption time, the MPC temperature setpoint as shown by series 3014 is shown to decrease to reduce an amount of electricity consumed by HVAC equipment.

Graph 3006 is shown to include a series 3018 illustrating power consumption (in kilowatts) of HVAC equipment over time. Similar to series 3014, series 3018 is shown to decrease at the start of the first peak consumption time in order to reduce an amount of electricity used during the peak consumption time. By pre-heating the zone before the peak consumption time, the predicted zone temperature shown by series 3012 is maintained above 78° F. (i.e., within comfortable bounds).

The control decisions shown FIG. 30 can be determined based on the deterministic model, the stochastic model, and the building thermal model determined from the data/calculations described above with reference to FIGS. 27-29. In this way, MPC can determine control decisions for the building/zone that optimize (e.g., reduce) costs and maintain occupant comfort even with heat disturbances affecting the building/zone.

Configuration of Exemplary Embodiments

Although the figures show a specific order of method steps, the order of the steps may differ from what is depicted. Also two or more steps can be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various connection steps, calculation steps, processing steps, comparison steps, and decision steps.

The construction and arrangement of the systems and methods as shown in the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.). For example, the position of elements can be reversed or otherwise varied and the nature or number of discrete elements or positions can be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps can be varied or re-sequenced according to alternative embodiments. Other substitutions, modifications, changes, and omissions can be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present disclosure.

As used herein, the term “circuit” may include hardware structured to execute the functions described herein. In some embodiments, each respective “circuit” may include machine-readable media for configuring the hardware to execute the functions described herein. The circuit may be embodied as one or more circuitry components including, but not limited to, processing circuitry, network interfaces, peripheral devices, input devices, output devices, sensors, etc. In some embodiments, a circuit may take the form of one or more analog circuits, electronic circuits (e.g., integrated circuits (IC), discrete circuits, system on a chip (SOCs) circuits, etc.), telecommunication circuits, hybrid circuits, and any other type of “circuit.” In this regard, the “circuit” may include any type of component for accomplishing or facilitating achievement of the operations described herein. For example, a circuit as described herein may include one or more transistors, logic gates (e.g., NAND, AND, NOR, OR, XOR, NOT, XNOR, etc.), resistors, multiplexers, registers, capacitors, inductors, diodes, wiring, and so on).

The “circuit” may also include one or more processors communicably coupled to one or more memory or memory devices. In this regard, the one or more processors may execute instructions stored in the memory or may execute instructions otherwise accessible to the one or more processors. In some embodiments, the one or more processors may be embodied in various ways. The one or more processors may be constructed in a manner sufficient to perform at least the operations described herein. In some embodiments, the one or more processors may be shared by multiple circuits (e.g., circuit A and circuit B may comprise or otherwise share the same processor which, in some example embodiments, may execute instructions stored, or otherwise accessed, via different areas of memory). Alternatively or additionally, the one or more processors may be structured to perform or otherwise execute certain operations independent of one or more co-processors. In other example embodiments, two or more processors may be coupled via a bus to enable independent, parallel, pipelined, or multi-threaded instruction execution. Each processor may be implemented as one or more general-purpose processors, application specific integrated circuits (ASICs), field programmable gate arrays (FPGAs), digital signal processors (DSPs), or other suitable electronic data processing components structured to execute instructions provided by memory. The one or more processors may take the form of a single core processor, multi-core processor (e.g., a dual core processor, triple core processor, quad core processor, etc.), microprocessor, etc. In some embodiments, the one or more processors may be external to the apparatus, for example the one or more processors may be a remote processor (e.g., a cloud based processor). Alternatively or additionally, the one or more processors may be internal and/or local to the apparatus. In this regard, a given circuit or components thereof may be disposed locally (e.g., as part of a local server, a local computing system, etc.) or remotely (e.g., as part of a remote server such as a cloud based server). To that end, a “circuit” as described herein may include components that are distributed across one or more locations. The present disclosure contemplates methods, systems and program products on any machine-readable media for accomplishing various operations. The embodiments of the present disclosure can be implemented using existing computer processors, or by a special purpose computer processor for an appropriate system, incorporated for this or another purpose, or by a hardwired system. Embodiments within the scope of the present disclosure include program products comprising machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine-readable media can be any available media that can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a general purpose or special purpose computer or other machine with a processor. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions include, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing machines to perform a certain function or group of functions. 

What is claimed is:
 1. An environmental control system for a building, the system comprising: heating, ventilation, or air conditioning (HVAC) equipment that operates to affect a variable state or condition of a zone of the building; and a controller comprising a processing circuit configured to: estimate a thermal resistance between air of the zone and air of an external space outside the zone using a plurality of values of a temperature of the air of the zone, a temperature of the air of the external space, and a heat transfer rate of the HVAC equipment that provides heating or cooling to the zone, each of the plurality of values corresponding to a different time step within a time period; for a plurality of time steps within the time period, use the thermal resistance between the air of the zone and the air of the external space, a time step specific value of the temperature of the air of the zone, a time step specific value of the temperature of the air of the external space, and a time step specific value of the heat transfer rate of the HVAC equipment to estimate a corresponding time step specific value of a heat disturbance; and operate the HVAC equipment over the time period using a model-based control technique based on the time step specific values of the heat disturbance.
 2. The system of claim 1, wherein the processing circuit is further configured to perform a system identification process based on the time step specific values of the heat disturbance to identify a predictive model, wherein the predictive model is used in the model-based control technique.
 3. The system of claim 2, wherein the system identification process comprises identifying a scaling parameter that scales the time step specific values of the heat disturbance.
 4. The system of claim 1, wherein the processing circuit is configured to identify a Kalman gain and a stochastic model of the heat disturbance based on the time step specific values of the heat disturbance or predicted heat disturbance values based on the time step specific values of the heat disturbance, the Kalman gain and the stochastic model used in the model-based control technique.
 5. The system of claim 4, wherein identifying the Kalman gain and the stochastic model comprises performing a multi-step system identification process.
 6. The system of claim 1, wherein the processing circuit is configured to filter the time step specific values of the heat disturbance through at least one of: an anti-spike filter; or a smoothing filter.
 7. The system of claim 1, wherein the processing circuit is configured to fit the time step specific values of the heat disturbance to at least one of: a Gaussian function; a sinusoid function; or a user-defined function.
 8. A method for operating heating, ventilation, or air conditioning (HVAC) equipment of a building, the method comprising: estimating a thermal resistance between air of a zone of the building and air of an external space outside the zone using a plurality of values of a temperature of the air of the zone, a temperature of the air of the external space, and a heat transfer rate of the HVAC equipment that provides heating or cooling to the zone, each of the plurality of values corresponding to a different time step within a time period; for a plurality of time steps within the time period, using the thermal resistance between the air of the zone and the air of the external space, a time step specific value of the temperature of the air of the zone, a time step specific value of the temperature of the air of the external space, and a time step specific value of the heat transfer rate of the HVAC equipment to estimate a corresponding time step specific value of a heat disturbance; and operating the HVAC equipment over the time period using a model-based control technique based on the time step specific values of the heat disturbance.
 9. The method of claim 8, further comprising performing a system identification process based on the time step specific values of the heat disturbance to identify a predictive model, wherein the predictive model is used in the model-based control technique.
 10. The method of claim 9, wherein the system identification process comprises identifying a scaling parameter that scales the time step specific values of the heat disturbance.
 11. The method of claim 8, further comprising identifying a Kalman gain and a stochastic model of the heat disturbance based on the time step specific values of the heat disturbance or predicted heat disturbance values based on the time step specific values of the heat disturbance, the Kalman gain and the stochastic model used in the model-based control technique.
 12. The method of claim 11, wherein identifying the Kalman gain and the stochastic model comprises performing a multi-step system identification process.
 13. The method of claim 8, further comprising filtering the time step specific values of the heat disturbance through at least one of: an anti-spike filter; or a smoothing filter.
 14. The method of claim 8, further comprising fitting the time step specific values of the heat disturbance to at least one of: a Gaussian function; a sinusoid function; or a user-defined function.
 15. A controller for operating heating, ventilation, or air conditioning (HVAC) equipment of a building, the controller comprising: one or more processors; and one or more non-transitory computer-readable media storing instructions that, when executed by the one or more processors, cause the one or more processors to perform operations comprising: estimating a thermal resistance between air of a zone of the building and air of an external space outside the zone using a plurality of values of a temperature of the air of the zone, a temperature of the air of the external space, and a heat transfer rate of the HVAC equipment that provides heating or cooling to the zone, each of the plurality of values corresponding to a different time step within a time period; for a plurality of time steps within the time period, using the thermal resistance between the air of the zone and the air of the external space, a time step specific value of the temperature of the air of the zone, a time step specific value of the temperature of the air of the external space, and a time step specific value of the heat transfer rate of the HVAC equipment to estimate a corresponding time step specific value of a heat disturbance; and operating the HVAC equipment over the time period using a model-based control technique based on the time step specific values of the heat disturbance.
 16. The controller of claim 15, the operations further comprising performing a system identification process based on the time step specific values of the heat disturbance to identify a predictive model, wherein the predictive model is used in the model-based control technique.
 17. The controller of claim 16, wherein the system identification process comprises identifying a scaling parameter that scales the time step specific values of the heat disturbance.
 18. The controller of claim 15, the operations further comprising performing a multi-step system identification process based on the time step specific values of the heat disturbance or predicted heat disturbance values based on the time step specific values of the heat disturbance to identify a Kalman gain and a stochastic model of the heat disturbance, the Kalman gain and the stochastic model used in the model-based control technique.
 19. The controller of claim 15, the operations further comprising filtering the time step specific values of the heat disturbance through at least one of: an anti-spike filter; or a smoothing filter.
 20. The controller of claim 15, the operations further comprising fitting the time step specific values of the heat disturbance to at least one of: a Gaussian function; a sinusoid function; or a user-defined function. 